Hi, Sarah! I lived in Hillsborough until fairly recently, so maybe I can help.
First of all, just wanted you to know that the town spells its name "Hillsboro
ugh". Unlike many towns around NC, Hillsborough has kept the silent "ugh" at the end of its name. It's a common mistake so don't worry about it, but when you're doing your hunting, it'll help you to have the correct spelling of the town's name.
As for utilitiy costs, that can vary a lot for an apartment. Many apartment complexes will include the costs of some utilities with your monthly payment, so be sure to ask what is & isn't included.
Water & sewer is handled by the Town of Hillsborough. Your bill can vary because of your usage, of course. But it can
also vary in Hillsborough depending upon whether you're living within Hillsborough-proper (which is a rather small area) or "outside" of town (and you may feel like you're inside of town, but the Town of Hillsborough may feel differently!). The prices for water are nearly double for people who live "outside" of town. But don't dispair. My husband and I lived just outside of town & we usually paid just under $30 a month for water.
Electric is handled by Duke Power (or are they calling themselves Duke Energy now? I forget). Monthly payments can vary a lot depending on the size of your place, how much you rely on air conditioning, etc. We were in a 2 story house so our payment can't compare to what you'd have in an apartment.
Cable is handled by Time Warner Cable. I'm not sure what they charge for basic cable. We had digital cable & high speed internet & paid about $100 a month, so you'll certainly pay less than that. Check their website for more info.
We didn't have gas at our place (heat pump instead of a furnace) and we use our mobile phones exclusively now, so I can't help you with the other two. But hopefully this will give you a start. Good luck!