Definition:

In most cases, instant means moment or in a moment.  There are three places where it does not mean that: Luke 23:23, Romans 12:12, and 2 Timothy 4:2.  The definitions for those are listed below.

In Luke 23:23, it means of the pressure of a violent tempest.

Luke 23:23 And they were instant with loud voices, requiring that he might be crucified. And the voices of them and of the chief priests prevailed.

1945 επικειμαι epikeimai ep-ik'-i-mahee
1) to lie upon or over, rest upon, be laid or placed upon
1a) on the burning coals
2) metaph.
2a) of things, of the pressure of a violent tempest
2b) of men, to press upon, to be urgent

In Romans 12:12, the term, continuing instant, means to be steadfastly attentive unto, to give unremitting care to a thing or to persevere and not to faint.

Romans 12:12 Rejoicing in hope; patient in tribulation; continuing instant in prayer;

4342 προσκαρτερεω proskartereo pros-kar-ter-eh'-o
1) to adhere to one, be his adherent, to be devoted or constant to one
2) to be steadfastly attentive unto, to give unremitting care to a thing
3) to continue all the time in a place
4) to persevere and not to faint
5) to show one's self courageous for
6) to be in constant readiness for one, wait on constantly

In 2 Timothy 4:2, Instant means be ready.

2 Timothy 4:2 Preach the word; be instant in season, out of season; reprove, rebuke, exhort with all longsuffering and doctrine.

2186 εφιστημι ephistemi ef-is'-tay-mee
1) to place at, place upon, place over
1a) to stand by, be present
1b) to stand over one, place one's self above
1b1) used esp. of persons coming upon one suddenly
1b1a) an angel, of the advent of angels
1b1b) of dreams
1b2) of evils coming upon one
1c) to be at hand
1c1) be ready,
1d) of time
1d1) to come upon
1d1a) of rain