Lessons you learned:
If you're within 20 miles of any significant weather, stuff will change.
He was joking it scared you, or scared him? Instructors have to let students experience the next level of things all the time. If it scared him, he needed to execute a go-around and/or tell you to do it and monitor.
It may have felt like an "ordeal" now, but you'll see worse if you keep flying long enough. Apply necessary control corrections and fly the airplane. Best advice I got as a student pilot was that as much as I wanted to apply smooth and slow control corrections, and have a "nice ride", there are times when a slow and partial deflection of a control surface simply isn't enough. The simplistic way of putting this is, "use everything you've got". If a wingtip is banking quickly and you don't want it to bank, rack the aileron to the stop if that's what it takes to stop the roll. You fly the plane, don't let the wind gust fly you. It takes practice in gusty conditions to get it.
Sure. Don't fly. I'm not completely kidding. Plenty of people limit their fun flying to totally calm days.
Here's the inherent problem with that, though... Eventually the forecast will be wrong. Someday there won't be a non-gusty airport within your remaining fuel distance.
It'll be a mistake in planning, but you'll still need to step up and perform and utilize the airplane to its maximum limits.
Maybe a single gust, maybe a never-ending pounding all the way around the pattern and down final with a landing made with the ailerons sometimes locked to the upwind stop as you react constantly.
You certainly won't WANT that to be how you arrive, but it'll eventually happen if you do this long enough.
You probably haven't gotten to the stage where the crusty old instructor asks you if you'd land ACROSS the runway if you had to?
In the end, you're responsible for the outcome of the flight. I've landed my STOL equipped Cessna once where the landing rollout was less than 100' without touching the brakes. Wasn't fun. Didn't have any other options and that runway was directly aligned with the wind.
Holding appropriate control deflections and taxiing it in was more of a scare than the landing itself. Others have had to go further and ask for wing walkers or just sit in it pointed into the wind and "fly" the airplane on the ground for a while until the wind abated. I haven't had to do that one yet.
You can kick yourself afterward for the bad planning. In the airplane it's about being in the moment and doing whatever is necessary to have a good outcome.
These are extreme examples. Your gust, sounds like you just needed to aggressively control the airplane. More on how a yoke affects that in a bit...
Aircraft dependent. Personally I'd rather be as slow as possible if I think I'm not going to have enough control authority to not hit something on the ground. Hurts a lot less. Force is a multiplier with speed. Break the airplane or break the humans, I'll take the airplane every time.
If you need more speed in a Cessna for flow over the tail, prop wash works as well as speed. Watch sometime when you get to the stage where you're flying in solid direct crosswinds and see what the nose does laterally when you pull off a gob of power in a slip into the wind. It'll go sideways downwind on you and you'll have to add more rudder. It's when the rudder pedal is already on the stop that you've run out of options other than adding power. The flaps on a Cessna aren't as involved as you think. It's the faster landing's time compression and a few seconds less WORK on the controls that creates a bit of an illusion that faster landings are better aerodynamically.
They're easier, but add more risk if the next gust takes you into the ditch and you hit something. You'll just hit harder. I'll take 10 more seconds of work in the flare for a lower rollout speed every time. And I'll take a go-around if I'm running out of control authority versus trying to salvage a mess on the rollout.
You'll know the limits when you hit them, because you'll literally hit the stops with the control wheel.
Let's talk about that yoke. Yokes make people behave like they're driving.
Sticks don't have a cross correlation to anything you already know.
With a yoke, people are timid in airplanes sometimes to roll the aileron hard over and quickly. That triggers a memory in their brain of how bad that would be in their car.
With a stick, if you need more aileron, you'll bruise the inside of your thigh smacking that thing over if you need it.
Recognize this mental game, and realize the airplane isn't your car. Full and immediate control deflection may sometimes be necessary and that yoke isn't a steering wheel.
Spin it over and hold it hard over if that's what conditions require.
Probably not from your description. Just pushing your personal boundaries. If it pushed the instructor's boundaries and you got to see how they handled it with a good outcome, consider it money well spent. That'll be your job soon, maybe with a passenger or two on board. (Maybe they'll even be barfing since you chose a really bad day to take them up. Hahaha. Try to avoid that. I've only ever had a single upchuck and it was someone I didn't think to ask how they do with things like carsickness and/or carnival rides. Turns out she couldn't even let someone else drive a car under perfect conditions without feeling ill. She took the airplane ride with her boyfriend to impress him. Haha. People. Sheesh. Me included. I was young and stupid.)
I think the option you missed and should consider is the go-around. Yes, if it's rolling and you need to rack it to the stop to stop the roll, that's one piece of it, but there's no need to continue after getting rocked or rattled.
Put the power up and climb back up and re-assess what's going on with the weather.
Take a minute to reconfigure your brain into "time to get more aggressive about aircraft control" mode and make a conscious decision that you're going back down for another attempt.
Or not. If not, fly to someplace more suitable to land. If you have the option. It's a conscious decision.
As long as the yoke isn't hard over or the wing isn't stalled, you still have options available. Don't be timid about stopping an attitude change you don't want. Put the attitude of the aircraft exactly where you want it and keep fighting to keep it there.*
From the description it sounds like a strong gust was just a surprise for you both. Various conditions can create them on even an otherwise calm day. After more time and practice in gusty conditions, the control responses will become more "automatic" but until then, you're consciously thinking about aircraft control and especially during initial training, you're supposed to be surprised by some conditions. That's why the instructor is sitting next to you.
If it makes you feel better, with decent weather planning and go/no-go decisions, I've only really been surprised by weather about ten times in 20 years of doing this. The story of a microburst I've shared in another thread, one VFR into IMC encounter, about five days where the gusts and wind came up and I felt unprepared but fought through it, and one bounce off of the ceiling with a loose lap belt on an otherwise completely clear and smooth day. Not even ten, really.
The bounce was probably the most insidious, in that it could have incapacitated me if it had been harder. I learned not to leave the belt that loose, evermore.
My head hurt for an hour, including the remaining 15 minutes to landing and the 10 minutes to taxi in. A David Clark headband dent in your noggin hurts.
* In severe turbulence, by the book definition of that, where structural damage may occur, you may have to accept more rocking and rolling and altitude gain or loss than you want. The above isn't about that, it's about simple gusts during approach.
