My relative inherited land back in India before 1980. India uses 1 April, 1981 as the cut-off date to figure out fair market value for inheritances that go way back in time. Can he use the same date for filing american tax? If not, is there any other cut-off date available? Is he free to choose his own date by any chance?

He also constructed an office on the land. However, he doesn't have any receipts for it. How would he prove construction cost?

Could you also shed some light on how indexing would work in this scenario?

For US tax purposes his cost basis is the fair market value of the land as of the date of death of the person he inherited it from - there is no other cut off date. He then adds the costs for capital improvements he's made over the years. If he doesn't have receipts he should attempt to reconstruct what he paid in order to come up with a reasonable cost basis, backed up with photos, canceled checks, and anything else that can help verify his numbers.

Indexing is irrelevant for US capital gains.

One more point to remember is that all items (cost basis, improvements etc.) have to be denominated in $$ using the exchange rate at that time. Even if they were not directly converted from dollars.