Transportation from the airport into Central Berlin

Take the train, it costs €4.

Use Google Maps to enter BER Berlin Brandenburg Airport as your starting point and your hotel as your destination. Choose one of the trains or buses. The S Bahn is extremely efficient and like the previous poster said, very cheap. ($200 shuttle?? Last year we took a FreeNow taxi to the airport from way over in City West and it was only sixty Euros.)
There are a couple of BVG ticket machines in the baggage claim area and one on the platforms. Make sure you buy an A-B-C zone ticket (the airport is in Zone C).

As far as what to do, if you don't want to go to a market, you can just walk down the KuDamm, which will be quite festively decorated. Lots and lots of people will be out, and last year at least there were Gluhwein and pretzel stands along the sidewalk. If you want to get a jump start on the markets, there is a market near the Kaufhaus des Westens, the big department store on the KuDamm, at the Kaiser Wilhelm Memorial. Without looking at the map I suspect your hotel won't be too far from there. Ask at the hotel desk for directions (a bus or UBahn will get you there quickly and easily if it's too far to walk) or ask for recommendations for another nearby market. If you think you might take the bus or Bahn somewhere that evening, buy a 24 hour ticket at the airport, then you won't have to worry about buying another ticket to travel again that day.

You can take either the RE 8 (every hour at 44 past the hour) or the RB 23 (every hour at 13 past the hour) to Zoologischer Garten station, then the 100 Bus to Luetzowerplatz Park, then walk about 100 yds to the Sheraton Grand. The S-9 also goes direct to Zoologischer Garten at 49 past the hour, taking a bit longer. Cost is 4 EUR.

Public transport is "cheapest and most efficient way".
Use journey planner to find best connections. Alternatively you can use BVG app.
Some basics about Berlin's punlic transport you will find here.

From BER to Berlin you will need ABC ticket, within Berlin city limits only AB ticket.
Do not forget to time stamp paper tickets for valisation.

Hint: The place's name in #3 is Lützowplatz; no "er". Lützow was a person, not a place. I know Lützowplatz well - my father was architect of the extension buildings from Hotel Berlin (now Berlin, Berlin).