Can I make money orders to pay an apartment?

Yes

You can get a cashier check from the bank

They sell money orders but I think the maximum amount is $500.00 so anything over that would have to be done in multiple money orders...for example $1000 would be two $500 money orders. Of course, you would have to check with the apartment management to verify that they accept money orders, but you can certainly buy them at Walmart.