It won't look as good without the new tires and wheels but why not. Plenty of people do it that way, just drive it with common sense and it'll be fine.
It will need an alignment after the lift is installed which is nothing more than adjusting the toe-in and straightening the steering wheel. Nothing else is adjustable, not the camber angle nor the caster angle (unless your lift has adjustable-length control arms) but neither of those will need adjusting anyway. So there is no need to pay anyone to do your alignment since you can easily and quite accurately set the toe-in and straighten the steering wheel with only a few common tools. Set your toe-in to 1/8" per the instructions at http://www.4x4xplor.com/alignment.html
and your alignment is done. No need to spend $60-80 for something you can do as accurately in just a few minutes.