There is a Youth Hostel in Takamatsu. See here:
http://www.hihostels.com/dba/hostels-Takamatsu---Takamatsu-Sakika-YGH-...Sadly, you can't book online, so you would have to call them, but it is about 4000 yen, according to the website.
I have a friend who has travelled all over Shikoku and I've spoken with her about it. Kochi was actually one of her favorite places (not just in Shikoku, but in all of Japan). The castle is one of Japan's 12 originals, along with Matsuyama's castle.
You listed 5 cities that you want to visit, right? I would probably do 1 day in each city. In Takamatsu, Ritsurin Garden and Yashima are great. That's the only city I have personally visited.
Travel between cities in Shikoku takes longer than most expect. From Takamatsu to Naruto, it is only 2 hours, but from Naruto to Kochi it takes about 4-5 hours. From Kochi to Matsuyama it takes about 4 hours.
If you cut something from your trip, I would personally cut Naruto. Going from Takamatsu to Kochi takes about 3 hours.