The jiu-jitsu academies listed here are in San Marcelino and Camins al Grau, and the fitboxing studio is in Ruzafa. None is inside Ciutat Vella, so expect a short metro, bus or bike ride from the old town.
Membership follows the gym pattern: a monthly cuota by direct debit, usually a matrícula on joining, and often a cheaper rate for two sessions a week rather than unlimited access. Ask whether open mat and the strength room are included in the price, whether there is a permanencia clause, and how to pay for the first month while your Spanish bank account is still pending.
On the mats, instruction is mostly Spanish but leans heavily on demonstration, and BJJ terminology is the same everywhere, so a beginner with no Spanish follows along more easily than expected.