If you are starting to get a urinary tract infection, can it go away on its own?

Yes. Drink plenty of fluids ( lemonade is particularly good) and void frequently.
Unlikely. Utis are caused by bacteria reaching the bladder and multiplying. So you need antibiotics to eliminate the cause of a uti. You can speed things up by drinking plenty of water, avoid any delays in urinating and treat constipation if present.