Armchair is used as an adjective (modifies "Analysis").
Analysis is a noun.
Sound is an adjective here (modifies "Armchair Analysis").
"Is this armchair analysis sound?"
Can also be written as....
"Is this amateur analysis reasonable?"
"Armchair Analysis" refers to an analysis coming from an onlooker observing from an armchair instead of someone actually involved in the field with real experience or expertise. "Sound" has many dictionary definitions related to whole, healthy, reasonable, fair, or good. I think of it as meaning "solid". I didn't find an etymology for it, but I believe it's related to the fact that broken items make a different sound when you hit them, especially if they are made of metal, glass or ceramics. If it has a nice sound you know it is sound (not broken).