Id personally prefer an a1 over a b1, but what are you looking to do with your car? That would determine what transmission you want.
But for future reference b18a1's and b18b1's are motors out of ls integras. These motors are non vtec, but have solid torque band if it is just going to be a daily driver. A b18c1 is what your were probably recommended which comes out of a gsr integra. It is vtec, creating quite a bit more horsepower, but just slightly less torque stock than an ls motor.
One thing you could do with your block if you are interested is put a b16a2 head or a gsr head on it and tap the oil lines etc.. to make it an lsv, but that requires more money and more knowledge so do your research.
Now back to the topic.
If you are just daily driving it, try to find an ls trans. It will keep you at lower rpm's on the highway, but if you just want to be a teenager and want to go fast and dont care about your mpg or the noise you make driving at 4k rpm's, go for a b16 tranny.
There is a solution for nearly everything out there in the world, just make sure you do your research before you make impulse and unknowledgeable decisions. It just takes one google search to find what u r looking for.
Give the noob a break guys, he obviously is new here :p