For now, it's silly: I just keep pressing F5 button to have a chance of being the one who answers the new question first. I know about RSS but would subscribing to RSS mean that I get RSS updated instantly when a new question is asked? In my Firefox, RSS feeds are updated every 30 minutes and setting it to 1 min or less would be insane.
In Firefox, I use the ReloadEvery add-on to set an automatic update time for the page to refresh, and set my Favorite Tags to the ones I want to see.
Just a heads up, but the front page and (I believe) certain other common tag pages are cached aggressively... last time I heard it's only rebuilt every 90 seconds or so.
I'm glad Lance's answer worked for you, but since this is Meta, here's the Stack Overflow/Stack Exchange answer to your question: the system does not support this feature.
Your only option is to use a third-party tool (or keep exercising that F5 key).
You can subscribe to StackExchange alerts for tags you want to follow which will email you new questions for a tag either ever 15 minutes, 3 hours or daily.