Berlin Marathon & sight seeing

I'm not an expert on Berlin, but I just returned last week so I can offer a little. Download the app for the trains, BVG. It's has an English option and was very, very helpful in planning routes, buying tickets, etc. When you come out of the airport, the lines are winding to the ticket machines for the train. If you have the app, it saves a lot of time. I think you can buy in advance and print if you don't have internet access immediately. I opted not to purchase the Welcome Card after mapping out my travel and what I wanted to see while I was there. As far as covering the airport to your hotel, there are different zones, A, B &C. Check where your hotel is located to see which zone to choose. If you plan on visiting museums, using the tourist bus, it might be a good option for you for discounts.
I did not notice vendors selling sim cards directly at the airport, but they were for sale at many of the stores in the city.
As far as shopping, you will be pleasantly surprised at all of the options you have for food. I don't think I've ever seen so many bio (organic) options anywhere. I don't think there is a Whole Foods there, but you won't miss it. So many great restaurants as well. You won't have to walk far to find something that will suit you. I was there during a marathon recently. What a great city, one of my favorites!

There are day passes for public transport that might prove more cost-effective than the Welcome Card, depending on your needs.

Be especially cautious about expecting to cover a lot of sights over just a few days. Berlin is a large city with spread-out sights. Public transport is efficient, but you'll spend quite a bit of time moving from place to place, which cuts into the number of places you're likely to have time for. And a lot of the sights are quite time-consuming (museums that need at least 2 hours for a short visit, 4-6 for a more comprehensive one).

I bought one of the cards (with Museuminseln access) and wished I hadn't, because I'd have had more flexibility if I hadn't been trying to cram card-related sites into just 3 days. And the card didn't save much, if any, money. The transit day passes I used later in my visit worked better for me. But it does depend on what you plan to do in the city. There may still be a discount on the Segway tour. If so, that might cover a good bit of the cost of the pass. And I didn't need airport transportation, which would also make a difference.

Prepaid SIM cards need to be activated first in Germany, this usually takes some hours. So it makes only little sense to do that at the airport. Can be done online or via phone. Don't rely on free WiFi, it's available only sparsely.

My experience with Wi-Fi was different from jjj's. It was in all my hotels in eastern Germany and most cafes and casual restaurants I went to. I don't know whether there are any city-sponsored outdoor hotspots, though.

In hotels it's pretty standard. At cafes you have to ask. But the OP has an apartment I think (because of own cooking) and I doubt there is any free WiFi available at the spot.

Thank you everyone for the feedback / responses. Very helpful.