The on-site service is usually called Remote Hands, and they do pretty much what it says on the tin. The ones I've seen are priced by hour.

If you do go on-site yourself, make sure to understand the DC's requirements for entry, and whether or not they provide a courtesy KVM-on-wheels so you can make sure you're able to diagnose your machine, assuming you haven't brought your own mini KVM or can use iLO/iDRAC/IPMI on a laptop.