Yes, this is possible if you are using a paid tier of Google Apps for Business or Education. You'll need to make sure you have it set up to manage devices, then set up your email account on your iPhone as an Exchange account. I recently did this to get back Push as well.
As you said, though, Google Apps does require you to have your own domain name, but that's as simple as buying one from any of the myriad registrars and then pointing the MX records to Gmail's servers.