Nothing wrong with any kind of bike, even a trendy "hipster" single-speed ... . but a single-speed might beat you up on those steep hills ... but since your identity isn't tied to your riding, you can simply get off and walk, like people used to (most "cyclists" would rather get hit by a car than push their bikes up a hill ... direct connection between ego and performance (I must confess to having a small dose of this disease myself.)
The British Racer you linked to has gearing, which is pretty handy in hilly terrain.
Another thing to consider is that you will need to wear a knapsack everywhere if you don't use a rack or basket. Either way works.
Nashbar.com might have some inexpensive fixie frames, as might BikesDirect. Your other best bet is local classifieds or Craigslist---not many factories sell the kind of bike nowadays, but there are a lot of lightweight, steel-framed English racers for sale used.