The conversion might just about be possible if you are very circumspect with your choice of components.
You should be able to get a suitable motor for about $130 but this will be a budget Taiwanese copy of the real thing. At this end of the market there isn't much difference in the price of the motor whatever displacement you go for. A smaller motor will take less fluid from the system but will deliver less torque and will stall more easily. Your friendly local budget hydraulic motor supplier might even let you try a motor out and then let you swap it for a different displacement unit if your first guess was too far out to be usable. Within any particular frame size each motor mounting arrangement is identical; as the displacement increases so does the motor length but the shaft size and front flange remain the same.
A couple of barrel type throttle-check valves will give you maximum speed limitation (one valve for each direction of rotation - use them to throttle the flow going IN to the motor) and these will cost you about $30 each. You will then need some hoses and couplings to match the tractor connections (I'm presuming you are using one of the auxiliary spools on the tractor's own hydraulic system for the forward/stop/reverse control). You probably haven't got enough money in the budget for niceties like anti-cavitation protection, but I dare say the auger doesn't have a huge inertia, it isn't going super fast and even the cheap motor is fairly robust and will manage without.
Then the rest of the money can be soaked up on the mechanical mounting for the motor, the shaft coupling, sundry hydraulic fittings and hose clamps, and, of course, your time.