If I push 160iso film by 2 stops, and shoot a scene at 1/500 @ f16 as if I would have shot it with 400iso film at 1/500 @ f16, will I get a similar exposure by pushing the 160iso film?
If your proper exposure using 100 speed film metered at 1/500 @ f16, and you wanted to push the film to 400 - then you need to change the exposure settings to what would be a proper meter using 400 speed film. In this case, 1/2000 @f16 (2 stops faster) changed the question to use 100 ISO to make the math easier
Obviously, this is under-exposing your film by 2 stops. You need to compensate for this by over-developing.
For example, Ilford Delta 100 shot at ISO 100 and processed in 1+4 DDX needs 12 minutes of development.
Delta 100 shot at ISO 400, also processed in 1+4 DDX needs 23 minutes of development.
This process, of intentionally underexposing film and overdeveloping, is push-processing.
The reverse can also be done (pull-processing).
Film and developer combos already have unique characteristics, and those change again when pushing or pulling, and more so depending on how much you push or pull a film. Some handle it well, others not so much. So, experiment away.
I would not develop myself unfortunately for the moment but it’s something I want to try with B&W in the future. Just got to find a good way to scan the negatives.
I would strongly encourage you to invest in some development supplies. All you really need is a Patterson tank and a change bag and you're set. Film/developer combos is a part of the fun of shooting B&W film. But, it's understandable that this isn't feasible all the time.
When using a lab, make sure you are absolutely clear that you need push processing done by X stops. Pro photo labs can do this easily. The local drug store, not so much.