Dissolving air in water using a bicycle pump

Using the home-made apparatus you mention you are unlikely to be able to generate enough pressure make a noticeable "fizz" like carbonated drinks. There are likely to be leaks. But yes if you increase the pressure of air in the container then the amount of gas which is dissolved in the water will increase (Henry's Law). Then if you release the lid suddenly you should see bubbles forming in the water.

Air saturation is similar to dissolving salt or sugar in water. For a given temperature and mass of water, there is a maximum mass of salt or sugar that you can dissolve. If you add any more salt or sugar it will just sit on the bottom of the container - you cannot get it to dissolve however much you stir it. At this point you have reached saturation.

Likewise if you bubble air or some other gas through a fixed mass of water at a fixed temperature and pressure, then only a fixed maximum mass of gas will be dissolved. At this point you have reached saturation. Any further gas which is bubbled in will come out again.

If the container is sealed then the pressure of the undissolved gas in the container increases as more air is pumped in. This increases the saturation point because it depends on the pressure of air above the water, so more air dissolves in the water. The saturation point is constantly changing because you are increasing the pressure. If you keep pumping then eventually you will reach a constant pressure at which air escapes out of the pump or the container as fast as you are pumping it in (or the container explodes). This limit is set by your apparatus, it is not a property of the water.