I came across this use of the word instant in the Dryden translation of Plutarch's life of Timoleon (who I must admit I'd never heard of before):

But through the care and diligence of his friends, who were very instant with him, and added force to their entreaties, he came to resolve and promise at last, that he would endure living, provided it might be in solitude, and remote from company; so that, quitting all civil transactions and commerce with the world, for a long while after his first retirement he never came into Corinth, but wandered up and down the fields, full of anxious and tormenting thoughts, and spent his time in desert places, at the farthest distance from society and human intercourse.

My dictionary does give "urgent, pressing" as one meaning of instant, but it's not a meaning I am familiar with.

Bingley

I have come across it occasionally, always used in that kind of situation (that is, '..his friends were very instant with him..'), and assumed, without checking up, that it had lost its battle with 'insistent'. I feel, though, that 'instant' implied an urgent insistence, usually pressing for some action before it's too late. A professional advisor may be insistent, but perhaps only a friend could get away with being 'instant'. So I think the usage had a value and rather regret its passing.

My dictionary gives "5 Archaic - presnt, current"
I have seen it used with dates given spelled out, e.g.
Friday instant meaning "this Friday".
I think the usage you quote would mean his friends were constantly with him.

I wondered about this: (so I looked it up! )

Main Entry: stance
Pronunciation: 'stan(t)s
Function: noun
Etymology: Middle French estance position, posture, stay, from (assumed) Vulgar Latin stantia, from Latin stant-, stans, present participle of stare to stand
Date: 14th century
1 chiefly Scottish a : STATION b : SITE
2 a : a way of standing or being placed : POSTURE b : intellectual or emotional attitude <took an antiwar stance>
3 a : the position of the feet of a golfer or batter preparatory to making a swing b : the position of both body and feet from which an athlete starts or operates


so, perhaps his friends stand with him.