República Hostel is so cute and the staff are so friendly. They were just starting out when I was there and Cesar, the manager, was very attentive and asking me what I thought of my experience there. They are working on new improvements every day so definitely put more weight on recent reviews. The building is beautiful with a gorgeous courtyard and pool. The towels were big fluffy and clean and only required a deposit. I was very impressed that they had installed key card readers into the really old doors. The drinks were very reasonably priced and the breakfasts were ample and included. Stephanie at the front desk is a gem! Other great things: wifi, beds, bed lights, daily activities, free walking tour, discount at local restaurant, and the rooftop terrace. I've also heard great things about their sister location in Bogotá.
Unfortunately, they only turn on the A/C in the rooms between 7pm and 8am. Cartagena is so hot that it is virtually impossible, for me at least, to be in any room without A/C or ventilation. After learning from Cesar, the manager, I found out that this policy is due to energy restrictions in the city and they will be fined if they use A/C during the day. While this makes sense, I would gladly have paid additional money to have A/C all day. The dorms were a little tight as it was basically one narrow hallway in between the bunks. Fortunately, the lockers are gigantic so I just stored all my stuff in there.
Even if you don't stay at República, you should definitely get a hotel or hostel in Old Town, not Getsemani. Staying inside the walled city is definitely worth the modest increase in price.