it all depends on the settings your camera has as to whether or not you can manually adjust the exposure time.
There are two things that you can play around with. Shutter speed and aperture. The shutter speed is what you are referring to when you speak of exposure time, it is how long the shutter stays open to let light in. Aperture is how wide the opening is. The smaller the aperture, the less light that goes in, which means you will need to leave the shutter open longer to make up for this lack of light. You can open the aperture up larger to let more light in, which will then allow you to increase the shutter speed to catch fast moving objects. The thing is, you need A LOT of light on the fish tank in order to get away with a fast shutter speed. You can pick up a 1000 watt halogen work light from Home Depot or Lowes for pretty cheap money.
On most tanks you won't have enough light to get really good shots without flash, but you can get good pics with the flash. A few tips for getting better pics with the flash is to put your focus on spot focus if it is available on your camera. This will help you focus on the fish and not the glass. Second, stand back a bit from the tank and zoom in as much as necessary. This should let you take pics from straight on without getting too much glare from the flash bouncing off of the glass.
If you have a lot of settings on your camera that you can adjust manually, just play around and see what works best. I use the Sony DSC-F828 and use it on manual mode. On the tanks that don't have enough light to go non-flash I usually have the aperture on 2.8 and the shutter speed around 500.