Congratulations on the trip! It sounds a lot like the one I took last summer, my two friends were broke so I went solo.
I'm not sure about alternative forms of transportation because I only used a eurail pass and a couple easyjet flights. Personally I opted for the eurail because I had no idea where I wanted to go so the flexibility worked well for me but if you have a more fixed itinerary it might be cheaper to go with something else. One thing though is that overnight trains are fantastic, you don't have to pay for a hostel and they are usually very comfy. When you go to Pompeii I would suggest booking your ticket a couple days in advance. I tried to go in early August and after 3 days of trying I still never managed to get there which is probubly the only regret of my trip.
I would say that booking a hostel in advance is a good idea but even in high season I found you didn't need much notice as a solo traveler. I never had a problem and was only booking 1-3 days in advance. I would simply book for my next destination once I decided and once I got there I would book the next one. If you go to Rome I would suggest you stay at the camping village roma i think it was called, it was on hostelworld.com and was one of the cheapest in Rome. It was super nice with 3 bed bungalows plus an on site bar/nightclub and 2 pools which were nice in the +40C weather. Plus it is a 5 minute walk from a supermarket which was nice, the one draw back is it a bit of a trek since it was a 45 min subway ride and then a 10 minute bus ride.
For your budget $1600 in 3 weeks would give you about $75 a day. I did Rome on about $50 a day so it is doable but with not many frills. As long as you plan on cooking quite a few meals and limiting some of the sites then you should be good. Personally I really enjoyed making my own walking tour, most hostels will supply you with a map so just catch public transit to the closest place and walk to the rest, you end up seeing a lot of the city, finding some cool little places and you can absorb the vibe of a place.
I hope this helps a little and have a great trip!
P.S. It might of been a total fluke but if you go to the Colosseum I would try going at about 12:30, I think everyone was hiding from the heat so I got in after only about 20 minutes in line. When I came by at 9am there was a line all the way around.
Clothes make the man. Naked people have little or no influence on society.
Places I've been: USA, England, France, Spain, Germany, Switzerland, Italy, Austria and Hungary
Next trip: Riding a motorcycle from Regina, Saskatchewan down to Central America, May 2012