Last month, I just moved into Manhattan from upstate NY (an hour away from NYC)where my family lives and started in my first job at the same time. In our employee portal, I changed my address from my upstate NY address to my NY City address so I don't have to go home too often to pick-up mail or have my family bring sensitive mail into the city every now and then. I also had most my bills mailed to my City address. When I got my first paycheck, I realized how much NYC taxes cost and that I could still change my address back to the upstate one on the Employee Portal and request not to have NYC taxes withheld. Is this going to be a problem for me even if I do have an address outside the city which I go to almost every weekend? I'm just starting work and I the money that goes to NYC tax could really help me in other ways.

Sorry, this won't fly. If you stay in NYC during the week - even if you spend all weekends upstate - NYC will claim that your residence is still in NYC. The only way to avoid the NYC residence tax is is to give up your NYC residence and commute to work each day from some place outside of the city limits (and outside of Yonkers as well). NYC is VERY AGGRESSIVE at tracking down people who work in NYC and have apartments in NYC but claim to not be residents of NYC.

If you work in NY state (even if you commute from another state), then the income must be reported on the NY state tax return.
You live in NYC, you must pay NYC taxes.
Your U.S. Tax Return: Working or Living in Two or More states

About the ONLY people who live in NYC, but who are not by law NYC residents are students, and THEY have to prove their student status EVERY year to be exempted.