You can as long as it’s a manual transmission. But push starting isn’t exactly easy and really requires a second set of hands to do safely. It’s also not advisable to do repeatedly as it can damage your engine. It’s mainly meant to be used as a one time emergency fix, not a long term solution.

Your best bet is to either find alternative transportation while waiting on a new starter or putting the old starter back in, giving it a few good whacks and hoping for the best.