Usually you'd want it the other way around lol (an H22 trans or something on an F23). But I guess if your A/T from the F23 is still going strong you could. Like Rusty said, ECU is gonna be a pain but it should fit.
You will care when you start driving it. You need to use an obd1 ecu to run the car properly. This means you need a jumper harness for the f20b or the H23. One does not exist for the f20b to my knowledge. I had to make my own for my current F20b swap and I had to make another one for my H23 swap. You need to identify the correct pins for the ECU such as VBSOL, TCS, D4ind, SHA, ATP1 & 2 and so on. If you dont know how to cross reference pins then this is already a bad idea. It would honestly be easier for you to swap in a manual. Its not cheaper obviously but whats the point of putting in a more expensive high performance engine and leaving on the bottleneck auto? You're better off putting in another F23. You clearly want to get better performance out of it so why bother.
Ulev is just Honda’s way of saying this engine is pretty much exactly the same as this one except it’s slightly more restrictive. I think there’s a negligible horsepower difference between the two. I think the way the ecu adjusts fuel trims is slightly different as the a4 is a little more restrictive. If you’re swapping motors you’d be fine with an a1.