*Besides,* as used here, is a preposition and **requires a noun** or noun phrase.

*Studying* is a gerund and functions as a noun. *Study* here is also a noun: you can test that by adding something like *more:* "besides more study for the test". 

All of [Oxford's examples](https://en.oxforddictionaries.com/definition/besides) follow *besides* with a noun (sometimes abstract, sometimes concrete; always a noun or noun phrase).

The question "which tense is right" is the wrong question, I'm afraid.