Can I use Music app without subscription?

I assume you are referring to doing this on your iPhone and not a computer (your forum and hardware selections indicate such)?

As long as you are using the same Apple ID your old purchases should appear unless the distributor pulled them from the store. You do not need an Apple Music subscription for that. Apple Music lets you sync via iCloud, plus it gives you access, on a subscription basis, to any music in Apple's collection.

At this stage it helps to know the exact hardware for which you are asking, plus system version information. There is a big transition in software and naming in the Apple environment right now. Apple certainly isn't helping by calling 3 different things "Apple Music". How you do things such as syncing are right now very hardware and software dependent, unless you want us to bombard you with 10 different links and leave you to sort out which ones you need. ;-)

I think the Music app is just old iTunes with the movie and podcast parts stripped out, then re-branded. :-) People are in a tizzy about iTunes "disappearing" but in fact people have been complaining about iTunes becoming bloatware for years now so I guess you can't win. Mind you, I think the real bloat has been Apple's media marketing components, and those are all still in there!