Can a hand drill drill 1/4"-1/2" holes into 1" thick aluminum?
I would say, "yes, but...".

First of all, you'll need the appropriate bit. Tungsten carbide is the go-to for most metal drilling, and such bits are usually also designed to drill efficiently into metal and remove the waste. You can probably also find diamond-tipped bits which will also be good at biting into the metal and removing it. Your average drill bit for hardwoods won't be up to the task; it won't bite, will dull quickly, and will bind easily.

Metal drilling usually requires use of an oil or other drilling fluid to cool the bit. This would make drilling by hand a two-person job; one to hold the drill steady, the other to apply the drilling fluid.

The last point of concern is not overstressing the drill's motor. This is usually just a matter of not pressing too hard, and letting the drill do the work in its sweet time. But, it also generally means having a drill with the chutzpah to handle the job. As was said in the comments there are plenty of hand drills with enough torque for this kind of work, but your $30 range is probably not in that class.