I realize I'm late to the game, and you already have something booked. But... if you are looking for a Rick Steves kind of place within your stated budget, I will throw out one idea that may or may not appeal to you: mittendrin. It's a 4-bedroom B&B in a flat in a building built in 1905 (as I recall) in the former West Berlin. It's located near Kaiser Wilhelm Memorial Church. It's owned by the lovely Sabine, a former stage actress in Munich. I've spent 50 nights there since 2015 and am adding 3 more nights in April. It's my favorite place to stay anywhere.
In your stated price range, there are two rooms that share a hall bathroom located between them -- one room is a single, the other can sleep 2. Sabine will happily clean the shower between guests. The small single room is typically 79-89 euro per night. The larger room that can sleep two is 89-99 euro per night (this is the room in which I stay). You mentioned you are looking for a room the last week of April. Not sure when you arrive, but I will admit, I have both rooms booked the nights of April 22-25 for me and a travel companion.
There is a nice common area where breakfast is served at a single long table. The breakfast spread is amazing and has a great deal of variety. As a frequent solo traveler, I enjoy the opportunity for conversation with Sabine and other guests at breakfast.
Sabine will do laundry for you if needed. She has never charged my anything for this. As for transportation, there is an entrance to a U2/U3 station at the end of the block. Two blocks over is the Wittenbergplatz U station (U1, U2, U3, and lots of buses). The Zoologischer Garten station (U/S/regional trains) is a 10-15 minute walk away.
No air conditioning, but that should not be an issue in April. There is an elevator, but you still have to walk up a half flight of stairs (two half-flights for the quickest path) to the B&B.
Neighborhood character? Hackescher Markt definitely wins. mittendrin is in a more commercial area -- hotels, department stores (KaDeWe is a block away), shopping complexes like Bikini Berlin and Europa Center -- but also sites like the Kaiser Wilhelm Memorial Church, Breitscheidplatz and the Berlin Zoo. With Berlin being so spread out, I spend most of my day out and about in Berlin, returning to the B&B to sleep at night.
Again, may or may not appeal to you. For many, the shared bathroom is a deal breaker, but I find it not to be a big deal. Half the time, there is no one in the other room. If you want to know more, PM me.