I dont think you will find anyone that will give you a server for free. Your better offering some sort of payment or even ask if you can rent the server per day. I remember seeing a few companies offering rental on servers per day basis.
Do you actually need a Quad Core CPU or would a 2x Dual Xeon be enough for you (meaning you still have four cores just divided on two CPUs) ? And maybe give some more detailed information about how much space you need (hard disk size) ?
Nevertheless expect it to cost at least $100 a month.
A lot of companies will do month to month. Depending on what you need, pricing will all be different. You're going to be hard pressed to find a day to day lease. Going for a month would be your best option