There are any number of commercial projectors which can be purchased which have an integrated scheduling system for powering on/off on a daily basis. This is not a common feature of residential projectors, just like your TV at home doesn't have that type of feature built in. Otherwise, you can use some type of control system and setup a schedule to power in on/off on a daily basis. I use Crestron and have setup schedules like this for clients who have had a similar need.

As far as a web browser, you would just connect a computer to the projector to do so. Something like a small NUC PC or similar and you just let it run all the time with the proper website setup and loaded and ready to go.