Should be possible for one skilled programmer to do what you want in less than a day. Bob will need to run a program that listens for requests from you to play certain files, then (if it plays nicely) will alert Bob that there is an incoming stream that he can then activate.
Time depends on the features you want to add. If it really is as simple as you say then it could be just a few hours' work to write and debug the code, since most of it is already written for you anyway (particularly if you use Java). If you want to create your own video player with lots of bells and whistles then of course you will need to push the budget and team size up accordingly.