Question: I'm new to using this scheduler. I will like to know if I need to have a web browser at the page so the scheduler's time always update? Or the scheduler will run so long it have been deployed to tomcat?

Answer: 
No, that's the whole point. The jobs run in the background while the server is running.
The scheduler requires no active connection to the server to be running (no open browser). In fact, if you are using quartz is mostly because you want to do some tasks on a time basis that does not require user interaction.