Can my landlord change the due date? And if so should the first month be prorated?

Since you are month to month, they can legally change the due date with the appropriate notice.

If you have no financial problem with paying on the first instead of the 12th, there is no point in arguing the issue, but the easiest method is as you suggest that your first new payment be for that part of the first month's rent not covered by the last payment and then on the first of each subsequent month you pay the full rent for that month. You do not ask and likely will be irritated by my bringing it up, but because I see the problem coming up so often, I suggest you make yourself aware of what your limiting legal rights are when there is no marriage [of for California, official "living together" status], as unless your name is on the lease, if you get into a fight with your boyfriend, he might try to evict you, and your joint earnings are not community property so you are not entitled to have of assets [only what you directly paid for]. The Michele/Lee Marvin case modifies this approach Nolo Press has some good books explaining the law and they are normally available at your local library.