i.e. Player needs 4 points to go out. The hand value is 3. Does the Player get to go out with the added 1 point "Go"?
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It's fine to go over in standard cribbage; the game is won by the first player or partnership to reach or exceed 121 points.
But remember the way things are scored: as they happen.
If that three points you mention was in play, such as opponent leads a 9, you play a 5, opponent plays a 6, and you play a 7, scoring three for the run in play, then the go from the opponent puts another 1 point score on the pile.
If it's a run in hand, however, it's scored only after the last card, starting with the player to the dealer's left and going left around the table (same as play order). If you are at 117 points, have a 3 point hand (say AJQK and a 2 up), and get the last card for 118, and the other guy is at 110 and gets 16 (AA23), it is determined by who scores hand first. If you dealt, you lose; if he dealt, you win. In three or four player non-partnership, if he's to your right, you've lost.
And remember: muggins can't be claimed until after the peg has been moved. Only on an overpeg would it prevent victory.
Also remember: if your scoring sheet shows last card separate from Go, there is no "go" on last card.
There is a common variant, called exact out, where you have to hit 121 exactly, and any score that would take one over is lost (but must be called out to prevent it being muggins-claimed).
A player pegs out at soon as they are able. You would score the points for playing the hand (Go's and anything else) first during the play of the hand, and then score your hands, non-dealer first. So in your case, since the Go was part of the play, you'd score the point, and then score your hand and peg out.
No you have to get the exact number to be able to win. If not you let the other person go until one of you gets the right number.