No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
There are quite a few rental management companies that are starting to report. This is meant to help those who rent and don't have a mortgage to report to their credit.
Rental history is one of the things that's been added to FICO 9, so if your apartment wants to report it, it can definitely go on your credit.
I wish my apartment would report, with full history of course. I've lived there over 25 years.