1560s, "a driving or impelling thrust," from push (v.). By 1590s as "a vigorous attempt." By 1803 as "a determined advance, a pushing forward." The sense of "persevering enterprise, a determined effort to get on" especially if inconsiderate of others is by 1855. Phrase when push comes to shove "when action must back up threats" is by 1936. An earlier Middle English noun push "a pustule, pimple, boil" probably is from pus by influence of push.
c. 1300, pushen, "to shove, move onward, strike with a thrusting motion, thrust forcibly against for the purpose of impelling," from Old French poulser (Modern French pousser), from Latin pulsare "to beat, strike, push," frequentative of pellere (past participle pulsus) "to push, drive, beat" (from PIE root *pel- (5) "to thrust, strike, drive").
Transitive meaning "urge, incite, press" is by 1570s; that of "promote, advance or extend by persistence or diligent effort" is from 1714; intransitive sense of "make one's way with force and persistence (against obstacles, etc.)" is by 1718. The meaning "approach a certain age" is from 1937. For palatization of -s-, OED compares brush (n.1); quash. Related: Pushed; pushing.
To push up daisies "be dead and buried" is from World War I, but variants with the same meaning date back to 1842.
"Pushing up the daisies now," said a soldier of his dead comrade. [The American Florist, vol. xlviii, March 31, 1917]
To push (someone) around "bully, browbeat, domineer" is by 1923. To push (one's) luck is from 1754. To push the envelope in the figurative sense is by late 1980s.
"in a position above and in contact with; in such a position as to be supported by;" also noting the goal to which some action is or has been directed; "about, concerning, regarding; in a position to cover;" as an adverb, "in or into a position in contact with and supported by the top or upper part of something; in or into place; in place for use or action; into movement or action; in operation," Old English on, unstressed variant of an "in, on, into," from Proto-Germanic *ana "on" (source also of Dutch aan, German an, Gothic ana "on, upon"), from PIE root *an- (1) "on" (source also of Avestan ana "on," Greek ana "on, upon," Latin an-, Old Church Slavonic na, Lithuanian nuo "down from").
Also used in Old English in many places where we now would use in. From 16c.-18c. (and still in northern England dialect) often reduced to o'. Phrase on to "aware" is from 1877.