Given a general sum game with the cost function
$A = \begin{bmatrix} 2000 & 0 & 2000 \\ 1000 & 100 & 1000 \end{bmatrix}$
$B = \begin{bmatrix} 400 & 0 & 0 \\ 1 & 1 & 400 \end{bmatrix}$
$A$ is the cost matrix for player 1. $B$ is the cost matrix for player 2.
Each player would like the lowest cost (think about the number as fuel usage)
My question is, would strategy pair row 1, column 2 constitute a nash equilibrium of the game?
According to the definition of nash equilibrium for general game, a strategy pair is a nash equilibrium if no player gains anything by deviating from his current strategy.
In our case, player 1 definitely do not wish to deviate from the lowest cost of the game. Player 2 may deviate to column 3 (to pick the other 0 fuel cost) but he has no incentive to do so.
But suppose I am player 1 and I pick row 1. But player 2 can pick column 2 or 3 yielding the same cost. Then I have a 50% 50% chance to either have the lowest cost in the game or the highest cost in the game. Would such a strategy violate the definition of nash equilibrium? Would such a strategy violate the no-regretness of nash equilibrium?
It's not quite clear to me what you mean by "such a strategy" in the last paragraph. You correctly quoted the definition of a Nash equilibrium, and the strategy pair "row $1$, column $2$" satisfies that definition. The fact that a player would be adversely affected by another player deviating from her strategy is irrelevant.