The tucking looks fine to me man.
Please, please, please, please, please stop benching in a shirt without side spotters or safety pins/rack! You are playing with fire. Even the best guys in shirts will throw one towards their face every once in a while. It will happen fast and it won't be pretty.
^^^^^ x2 no side spotters is never a good thing.
That bench looks very high...Your feet are barely touching the floor. If you're competing in a fed that requires you have feet flat, you should be practicing driving your heels into the floor hard. Otherwise, try bringing your feet back and pushing through on your toes which should help with your arch. Either way, put some plates on the floor if you can't touch the floor. Also, keep working on your tightness, your entire body should be tight (especially the shoulder blades pulled back and together).
Makin' a lot of progress bro! Now that you're touching in the shirt time to fine tune your form... Be patient and it'll come.
In addition to what the two previous post said, one thing I noticed is you're not holding the bar at the top after completing the press. Throwing the bar back after the rep is completed is a bad habit I used to have too...Hold the weight at the top for a second just as you would during compettion and have your training partner give the "rack" command.