How long does one bacterial cell live?

Bacteria are essentially immortal. That is, there is no limit on the number of divisions each cell can undergo. This is in contrast with multi-cellular organisms, where cells typically undergo a programed number of divisions, then stop or sometimes die off. You do have constantly dividing cells in your body, like in the lower layers of you skin and other epithelial tissues.

You can think about it in terms of strategy for survival. In multi-cellular organisms, survival is dependent on attaining a particular shape, to swing through trees or eat grass or photosynthesize or what have you, which requires a carefully coordinated growth pattern. In bacteria, and other single-celled organisms, cell division does't need to be limited in that way. There are other limitations, like intra-community chemical signalling, but each bacterial cell is normally capable of undergoing a theoretically unlimited number of divisions for as long as there is food available, and definitely don't "age" in the sense you mean.