So far as that goes, and being as you just bought the car...

I would first be sure the engine cooling system is working as it should.

Then check out the cold starting issue as well as fuel pressures and internal leakage.

Then next I would do a general tune-up, replace filters, fuel filter, air filter, oil filter, change oil, replace spark plugs* and distributor cap/rotor.

*And BE DARN sure to use the proper spark plugs as specified for your car. My car requires the non-resistor type and is much happier with those. Don't listen to auto parts guys and note the previous owner may have installed the wrong plugs!

Search on the internet forums for this. Find what people have to say by including the word "forum". Like

forum Mercedes W126 spark plugs

Using the word forum will weed out all the auto parts stores selling spark plugs, etc.

W126 is another name for the model years of your car...
https://en.wikipedia.org/wiki/Mercedes-Benz_W126

(Again, read, read, read.)

If you still have the idle up/down issue after the above, THEN start troubleshooting other things for that problem.