outdo v.

in Old English a common prefix with nouns, adjectives, adverbs, and verbs, "out, outward, outer; forth, away," from out (adv.). The use was even more common in Middle English, and also with the senses "outer, outside, on the outside, from without, external, externally; apart; greatly, extremely; completely, thoroughly, to completion." Other senses of out that extended into the use as a prefix include "beyond the surface or limits; to the utmost degree; to an explicit resolution."

In composition out has either its ordinary adverbial sense, as in outcast, outcome, outlook, etc., or a prepositional force, as in outdoors, or forms transitive verbs denoting a going beyond or surpassing of the object of the verb, in doing the act expressed by the word to which it is prefixed, as in outrun, outshine, outvenom, etc. In the last use especially out may be used with almost any noun or verb. [Century Dictionary]

"perform, execute, achieve, carry out, bring to pass by procedure of any kind," etc., Middle English do, first person singular of Old English don "make, act, perform, cause; to put, to place," from West Germanic *doanan (source also of Old Saxon duan, Old Frisian dwa, Dutch doen, Old High German tuon, German tun), from PIE root *dhe- "to set, put, place."

Use as an auxiliary began in Middle English. Sense of "to put, place, lay" is obsolete except in phrases such as do away with. Periphrastic form in negative sentences (They did not think) replaced the Old English negative particles (Hie ne wendon).

Meaning "visit as a tourist" is from 1817. In old slang it meant "to hoax, cheat, swindle" (1640s). Slang meaning "to do the sex act with or to" is from 1913.

Slang do in "bring disaster upon, kill" is by 1905. To have to do with "have concern or connection with" is from late 13c. To do without "dispense with" is from 1713.  Expression do or die indicating determination to succeed despite dangers or obstacles is attested from 1620s.

Compare does, did, done.