You can't connect Microsoft Project 2013 to Project Server 2010, no.
I don't know how O365 updates are planned, but Microsoft Project Server 2010 and Microsoft Project Professional 2010 are both no longer eligible for mainstream support since 2015 (meaning only security updates are released for these products, unless you pay Microsoft for a hotfix). Extended Support will end in 2020, after that point no support will be available. Since these things can sometimes take a while to be approved and budgeted, I would advise you to start planning for a migration.
https://support.microsoft.com/en-us/lifecycle/search?alpha=Microsoft%20Project%202010