You may find complete avoidance of public transport a little difficult if you want do a lot in Rome in a short time. While it's an entirely foot-friendly city, you may want to make use of buses, trams or metro depending on preference. We're long-distance walkers and not bus/tram people but have hopped the metro just a couple of times over our trips when wanting to save some time between longer distances.
Rick's guide or any number of others (check your public library) will be really helpful, and some reading is advised before you go. And take a look at hotels in your price range at booking.com or Venere.com. You will be able to easily see the big cluster of them located in the central area on their maps, and check reviews/ratings.
"Central" is sort of a sketchy term but I'd loosely define it as west of Termini, east of the river, south of Villa Borghese, and north of the river's southern-most bend and Colosseum. Then again, we've stayed up in the Prati neighborhood, - northeast of the Vatican - enjoyed it very much, and walked everywhere from there although it is a hike to the Forum/Colosseum/Palatine and other things down that way.
The time it takes from one spot to another is difficult to figure as everyone walks at a different pace, and most folks need a little extra time with a map to figure out where they're at (very few straight streets in Rome). We also find so much to look at along the way that it can take us a LOT longer what with all the dawdling!