im guessing the blinds were 100 200? so if so i would have raised 600 to 700 UTG. i dont like min raises. If the blinds were less then the raise might have been okay. Your c bet the flop was good amount. then making him pay for the str8 or flush draw with all in shove on the turn was good. I think he played it very well checking his set and letting you bet. You just ran into a better hand. however a lot of people will tell you dont risk your entire stack on just top pair. I think a lot of people would have played it like you did. maybe a raise of 600 to 700 preflop would have got the 44 to fold we will never know.
It's hard to back off on a hand as good-looking as AK. My little poker ship has run aground on the rocks with AK more than once. I now do my best to take it slow.
In this hand I couldn't see your pre-flop bets but I assume they were on the min side. Your post flop bet was a little high for my liking but once he called you should have slowed way down. Your opponent's range would certainly include Ax so you should have been wary of not just a set but also two pair. It's hard to see AK as being the worse hand after the flop but after the turn you should have considered the fact that all 4 cards on the board hit your opponent's range. Once he called your somewhat high flop bet your alarm bell should have been ringing. I would have checked here. Your opponent might have checked back to feign weakness hoping to trap you on the river. After that you're check-folding.
I think you had to raise depending on your opponents. Sometimes you can't do a standard raise because you can get more than 2 callers. Depending on the table, the effective stack and your image you have to raise really strong. 6x or more.