Any way to find out how much my competitior is charging on the same route?
No.
You could send him a PM and ask politely.
But pricing isn't this important and increase prices or decrease prices makes no big differences in most cases.
Then what is important? If my competitior already has an established route and has taken up all the route demands, does it mean that I can't fly that route?
Just fly the route.
Important are things like used aircraft and how old it is, how often you fly the route (daily and how often a day), what time you fly it, Company Image and Route Image.
There are lots of variables and price is just one.