Am I suppose to pay sales tax on capital improvements on my home in NY

Yes, you will pay sales tax on goods you purchase for home even when it is capital improvement.

If the sales tax was charged, you can pretty much assume that the sales tax was due.

Believe me, if the purchase was exempt from such taxes, the vendor would use the exemption. Sales tax collection is a REAL adminsitrative burden to the vendor which he will forego if at all possible.