I live in an apartment and the water shut off valve is in the apt next to mine; in order to reach the valve, we would have to break through dry wall.:eek:

Unless you have any shut-off valves near the source of the leaks, you may just have to go through the drywall.

There is also the possibility that there may be a general shut-off valve in the basement, whereas, you wouldn't need to go through drywall? Although, using this valve might affect other apartments?

One of the only ways to contain a pressure leak without shutting off the water would be to use a saddle clamp,(see image). I have heard about freezing the pipes but have never tried it. Regards, Tom

There are different ways, including the freeze packs. Post back with details, maybe a picture.

Depends a lot on how much room you have to work with and the type of leak.
If it is just fixing a hole on the side of a pipe a clamp similar to what speedball suggested works great.
Freezing kits generally don't work for leaks as the water will not freeze if it is running.
They are good for changing valves or fittings that are not leaking.
If you live in an apartment call the super or the owner. I have had tenants try to fix leaks when they didn't know how. It took quite a bit of my time to repair the extra damage. Most landlords will repairs leaks quickly as the damage and costs of water add up quickly.
If you describe the leak a little better it will be easier to give you an answer.