For completeness: If Alice chooses yellow, Bob chooses the nearest green and gets 12 points.

This is particularly great because the same number can be placed as an exponent on each of the larger numbers to make it work. When I opened the first spoiler, I thought, "Yeah, it's the reciprocal of the accepted answer and can be used in the same way," ...and then I opened the second spoiler.

@sayanel - In the case of odd-sized boards, first player has a winning strategy. Their first move plays in the center, subsequent moves answer second player's previous move using the same strategy as this answer.

@FirstNameLastName - You're right. Fixed.

@FirstNameLastName - I wrote a C# program. I will add the code to my answer.

According to my Monte-Carlo simulations, this is probably incorrect. The average score for Bob given 39 1's and the strategy you describe is 62.47, with a margin of error of 0.006 The Monte Carlo seems to say that 42 1's is the best Alice can do, (with Bob scoring 59.99 and a 0.006 margin of error), but I don't know how to prove it mathematically.

@emanresuA - The difference is that your nemesis knows how far they start from the exit. This puzzle is to either figure out how they can use that information, or prove that that information doesn't help them.