Actually, the isopropanol+toothbrush method isn't really necessary in this case. It's great for removing flux residues after soldering, because this type of residue doesn't dissolve well in water, so you need both a solvent and some force to get it off.
What I would recommend, unless you live in an area with hard water (ie mineral rich water) is to open up the cartridge, take out the battery, rinse it in tap water, and leave it to dry.
Note that removing the battery erases any save data, so make sure you back it up before doing this.
The crucial part is getting it dry. Something like rice or silica gel bags can be useful here.
Additional information:
There are three different ways electronics can fail due to water.
1) The component itself is sensitive to moisture, typically because it contains something that can react with water. For a GB cartridge, this would be the battery, which contains lithium, and the electrolytic capacitors, which contain an aluminum electrolyte. They will probably both be sealed well enough that a quick rinse will not destroy them, but at least you can remove the battery so you probably should. Other than that, the components do not get damaged by short exposure to water.
2) Shorting. If the device is powered while exposed to water, especially a liquid with a lot of electrolytes like, uhm, beer, two points in the ciruit might be shorted and cause damage. Unless something is actually damaged while the liquid is still on the circuit, this is not fatal.
3) Corrosion. The traces on the circuit board, the solder and the component leads are obviously made out of metal and might corrode ("rust"). This process can be accelerated if any kind of electrolytes are present. Not that simply leaving the PCB to dry is not enough if there's residue left, as will almost certainly be the case with the beer. Moisture will be picked from the air and slowly corrode whatever components are covered in the gunk. You should absolutely clean this up one way or the other.
As for point 3, hard tap water, and even soft tap water will leave some amount of residue. I wouldn't worry about that unless you can see a visible amount of residue when the water dries up. If you can get it or make it, distilled water is even better. Making it could be a fun experiment. If you're still in school, they probably have bottles of distilled water in the chemistry halls.
Good luck.