Don’t worry, it is all very accessible and public transport is great. Betahaus is just off the U8 subway stop “Moritzplatz”.
In Mitte, you have the well-known St. Oberholz, a café with adjacent coworking space right at Rosentalerplatz. A few weeks ago, they opened a second space 5 min away. If you prefer it a bit quieter, across the street of St. Oberholz is Fabisch/Circus Cafe.
Close to Friedrichstraße is the “Basecamp”, a cafe/coworking space from Telefonica. You can just show up and get a coffee and work. There are also more upscale coworking spaces which usually require a monthly membership, There’s WeWork at Potsadamer Platz and a new one at Hackescher Markt. There’s Mindspace close to “Stadtmitte”. Those are the ones I visited so far, you’ll easily find more.