If your not taking time away from something important to learn the language then go for it. Otherwise it won't make a huge difference from going in with no Mandarin.

Most western hotels will have English speaking staff who can help you with points of interest and writing down places to show cab drivers. While in the city it shouldn't be very difficult to find someone who speaks English as long as your not shy about asking for help. The thing is most younger Chinese people (under 40) have some English ability but are simply very shy about their ability so may prefer not to engage with you. On the other hand many young people would love a chance to practice their English so as long as your within 4th Ring Road in Beijing, finding someone who can speak English isn't impossible.

As for Shanghai, I would say English speakers are even more abundant so as long as your in the city, you shouldn't have too much trouble (especially with a smartphone in hand).

The only times where Mandarin would be a must is if your going to try to take public buses or hope to not get ripped off in places where you traditionally haggle for prices. With your condensed schedule I doubt you'll be going too far out of the cities anyways (aside from tourist spots), so wherever you'll be should have English speakers (whether they be University students or tour guides).

Taking a few weeks of the course may help you become familiar with how Chinese people interact along with some basic phrases though. The courses won't make a world of difference but could be interesting nevertheless.

Good luck and I hope you have a great trip!