There are two white areas on the right side of my monitor. The monitor is pretty old, a 15" Dell 1501P analog, and I've had it for 7 or 8 years
Is this a sign that the monitor is nearing its end? I'd like to get a new monitor, but I don't want to until this one completely dies. I've attached a picture of the spots.

Dimming, limited control of contrast, and limited control of color are typical of an old, worn monitor.

Your grapgic is more like an artifact - typical for a gpu problem. Does this bright spot follow the black border when adjusting the monitor horizontal size/position? Move vertical position and observe its effect.

Those 2 spots may be associated with the horizontal wires securing the shadow mask for color monitors. I know nothing about such things. With the gpu card trying to put a border there (black), it is hard to envision what electrical or mechanical condition is revealing itself as bright spots.

I do not know how to move the vertical position, but i rotated the display under the nvidia control panel and the spots stayed in the same place(in comparison to the physical monitor, not everything on the screen).

I would say the LCD panel is bad

Thanks. Any ideas how long it might take for it to become completely unusable?

LCDs can develop dead pixels over time. But you can try free dead pixel programs to try and revive them. If it is a large area, it could be something else.