What is the best Rear Derailleur for single speed to 3 speed conversion
Yes - Altus are a step up from the Tourney lineup, but is considered a very entry level MTB product. I would encourage you to consider a higher spec model if you can afford it - I suspect the price difference will be small if you shop around. 

Derailleurs are rated by the capacity and maximum tooth size. As you are run a x1 setup, you capacity requirement is 22-16 = 6, and the maximum tooth size of 22 is smaller than a typical road cluster, so any derailleur will do the job. As such, using a short cage will give better performance. 

You don't say what spacing the three cogs are - If you are using an indexed shifter make sure it matches the cog spacing of your cluster. As always Sheldon Brown has all the info you need