Where is your AC distribution panel located? Rather than run a circuit from the existing receptacle location, run the circuit from the breaker panel itself especially if it is closer. As long as you have expansion space in your breaker panel, it is also very easy to add additional 220 breakers so you could have a dedicated circuit for both the compressor and the welder.
You can do what you propose very easily but you will need to use emt or rigid conduit and weatherproof fittings since it would be on the exterior. If you can run all of the circuit inside the house by going from the breaker panel you can useemt, pvc or even direct install without conduit.
The shorter the run of the circuit, the smaller the gauge of wire you can use. You will find the cost of wire per foot shoots up when you go from say 12ga to 10ga. It is like long thin sections of gold.
I've done a lot of wiring like this and I know how to do it so it is code compliant, if you need any help let me know.
I would have to look it up to be sure but I believe for a 50a circuit you need to use a 3+1 conductor 10ga wire when the run is 100' or less. Keep in mind too that 50a is probably way more than what you need for a compressor or newer welder. Usually a 30a circuit at 220v is more than enough. This can make a significant difference in the size/cost of the wire you use.
Randy Rosetta KDØKWX TLCA #19351 69 FJ40 "Meanie" & 1970 Boss 302 Mustang