Watts are watts regardless of the source. Power in watts = Current times voltage. So power divided by voltage = current. 700W divided by 12VDC = 58 amps. Modern inverters are getting pretty efficient. Lightly loaded ones are less efficient. At full load, a modern inverter is probably better than 90% efficient. But that efficiency also needs to be taken into account, so instead of 58A to make 700W, you need closer to 65A to make 700W thru the inverter. At 12V, this ammount of current also takes really large wire.
Unless your alternator is greater than 60A output, it will see this load like a short, and do it's best to fill the hole this size load creates. The battery will buffer this load to some extent, but ultimately the energy has to come from somewhere. Alternators run beyond their max capacity don't live long.
Drill motors are inductive devices. Inductive devices draw the majority of their energy from the peak of the AC sine wave. Unless it is a high end inverter, it is probably a modified sine wave type, or worse, a square wave type. A modified sine wave is a series of square wave signals stacked to resemble a sine wave. But since it is not an actual sine wave, the drill will be less effiient and draw more power to perform the same ammount of work...
What does a new alternator cost, plus labor to install it? I would guess that you can get a real nice cordless drill and a few extra batteries for that same cost. You will also not have to worry about cords and such, as well as properly wiring a 12V inverter to your tractor. The drill will also be usefull elsewhere. Once you have the batteries and charger, you can also add tools that use the same battery packs. I have a 19V sawsall that I am really impressed with. I even put it on a pole, and use it to limb trees...
Good Luck.