Do universities typically cover publication fees associated with open access publications?
Talk to your professor who will know better than us. It's possible your professor has a grant that also has an open access component, in which case that's where the money will come from. It's also possible your university's library has set aside some funds for open access. Here's an example page from the University of Hull. If your university has something similar, you might be able to get funding from there.

If you find you don't have funding, you can pay the OA fee out of your own pocket (probably not a good idea even if you are wealthy), decline OA (if it's a hybrid journal which publishes both OA and non-OA papers), ask for a waiver (Elsevier might grant this since you are from India, which is a developing country), and finally withdraw and submit elsewhere. If you take the last option, do so quickly so as not to waste the journal's time.