by 1914 (1903 as push), a word of uncertain origin, but there is no evidence for the common derivation from an acronym of port outward, starboard home, supposedly the shipboard accommodations of wealthy British traveling to India on the P & O Lines (to keep their cabins out of the sun); as per OED, see objections outlined in G. Chowdharay-Best in Mariner's Mirror, January 1971; also see here. The acronym story dates from 1955.
More likely it is from slang posh "a dandy" (1890), from thieves' slang meaning "money" (1830), originally "coin of small value, halfpenny," possibly from Romany posh "half" [Barnhart].
The cavalryman, far more than the infantryman, makes a point of wearing "posh" clothing on every possible occasion — "posh" being a term used to designate superior clothing, or articles of attire other than those issued by and strictly conforming to the regulations. [E. Charles Vivian, "The British Army From Within," London, 1914]