I believe it was a generic Danfoss compressor, but it would depend on the age as they used 2 different refrigerants for the 1800.

CM1800-3

R404A 

I believe that should be the Danfoss SC15G 120V(?)  this should be available from a local refrigeration supply house.  There should be a label on the compressor in the unit that you can verify this info.  

Also, I am not sure what the issue is on the unit, but I have found that there are a lot of things besides the compressor failing that will cause it not to cool, so if you compressor is running and you have ok refrigeration pressure, then it could be something else that is the issue.

115V compressor Secop Danfoss SC10CL 104L1503 R404a/R507 refrigeration

THE COMPRESSOR IS MISSING. Can I use this compressor?

no, that is the wrong one it should be a 

Secop SC15G R134a Hermetic Refrigeration Compressor 1/2 HP

also confirm the voltage, some versions were 220v.  The compressor says on it that it is for R134A, but use R404A.  Feel free to email me if you need further assistance.

The cryo tag indicated 120V 60Hz R404A.

SORRY TO BOTHER YOU.

IS THIS GOING TO WORK OK:

Secop NF11FX R134a Hermetic Refrigeration Compressor 1101 BTU LBP 4164 BTU HBP

Secop Compressor Model NF11FX-2

Refrigerant Type R134a

1250 BTU/Hr. -  LBP -10F

4840 BTU/Hr. - HBP +45F

115-127V/60Hz

sorry, that is different spec, so probably not.