How can a 12-year-old boy earn 350 dollars in under 5 months?

All kinds of things. That's not a whole lot of money so it's very possible at that age. You can't get a job legally, (work for an employer), so you could get by with odd jobs here and there. You could mow loans, and charge $10 per lawn, sell lemonade or candy, etc. Also, ask your parents to see if they could pay you for extra chores around the house.