Some banks might prepare one for you IF you gave them cash; but they will charge you. Most big banks will charge you at least $10 for this service. Ask the landlord if they will accept a money order, which can often cost under $1. You can get them at US post offices, Western Union, SOME banks, and some retailers like Wal-Mart ($0.60).
I don't see why you wouldn't be able to pay cash for a cashier's check. Just call a bank and ask.
find out if they'll take a postal money order and if so get one or more of those. otherwise have someone you know who does have a bank account purchase the cashier's check for you