Naruto might also gain better chakra control. Hashirama was known for his vast chakra reserves and precise control. Under his training, Naruto could refine his own chakra control, which would make his existing jutsu more efficient and powerful. For example, his Rasengan could become more stable and have greater impact.
Another potential ability is an improved connection with nature. Hashirama had a deep bond with nature. Naruto could learn to sense the natural energy around him more acutely and use it in a more refined way. This could lead to new forms of Sage Mode or enhanced versions of his existing nature - related jutsu.
He could potentially gain an enhanced understanding of wood - style techniques. Since Hashirama was a master of it, Naruto might learn how to create more complex and powerful wood constructs, like huge forests for both offense and defense.