This was a great puzzle, and I received correct answers from Suman Saraf and Prakhar – very well done!
Please see the figure below.
Let’s divide the entire board, except for the central square, into 2×1 dominos. For example, it can be done as shown in the diagram. At each move, Anisha shifts the marker into some square of the same domino. If Arushi then moves the marker to the second square of the same domino, she will always be able to move, and thus Arushi will win.
Hope you all enjoyed the puzzle!