Berlin lies in the Northern German lowlands, which are flat as a pancake. To find a hike with a elevation change of 1000ft you first have to travel for about 3 hours either to the west or south.
Hiking in the Spreewald is beautiful, but it's something completely different. Forrests, meadows and waterways, cute little villages, if you are lucky kingfishers or dippers, but no grand vistas.
Is there a map anywhere of the hiking trails? Are the hiking trails
the hilly-dirt-trail-through-the-woods kind or are they the
Typical hiking trail
Hiking maps exist of course, but they usually also show cycle paths and waterways. And for just one day I surely wouldn't buy one. The area is very touristy and well-marked, and printing out a online map, or a offline map on a smartphone, should be enough. You would probably start in Lübbenau, and the most popular hikes from there are the Sorbian villages of Lehde and Leipe, and a popular restaurant called Wotschofska, which are almost impossible to miss.
But I would recommend something different: how about Potsdam? Potsdam is surrounded by beautiful parks and lakes. Most foreign tourists usually just visit Sanssouci palace, but the park is HUGE. If you want to see everything... New Palace, Chinese Tea House, Roman Baths, Ruin mountain, Belvedere and much more.... you need a full day, and you would walk ~10km or so.
Here is a list of all structures in the park:
Buildings in Sanssouci Park
Or see the other parks of the city. Last year I did this walking tour:
From Babelsberg to Belvedere on Pfingsberg and back
Take the bus from Berlin to Glienicke Palace and cross the famous Spy Bridge. Next is the northern end of the Berliner Vorstadt, the poshest part of the city where Germanys rich and famous live, with a reconstructed royal pier called "Kongnaes". Then you cross the New Garden with Cicilienhof Palace and hike up to Belvedere on the Pfingstberg. From there back to the New Garden, via the Humboldt bridge to Babelsberg Park with Babelsberg Palace . Before you return to Glienicke Palace you cross a cute little village with Swiss cottages, a hunting castle and a very good ice cream parlor.