Unfortunately many of your questions cannot be answered because of the lack of information. It is inlikely however that your system is as old as the house.

First, a cesspool is a pit in the ground which received sewage and water. There is no septic tank before the pit. These are the hardest systems to treat because of the quick buildup of anaerobic conditions in the pit which speeds the clogging of the pit. 

The first thing you should do is to determine what kind of system you have. If you have a septic tank, have it pumped and inspected. If you have a cesspool of a seepage pit, have it opened,inspectedn and cleaned. There are things that probably should have been done before closing on the house, and certainly need to be done now. Putting this off may lead to a significant expense of system replacement. It is possible that some of the yard will need to be explored in order to do this. 

The expense of doing this will vary depending on where you live.

If you have any questions, please call me to discuss your particular situation.