My spreadsheet displays fractions in 64ths. Ex. 53/64 th. Question: How do I round and display this number to the nearest 1/16th? In this case 53/64 would round to 13/16 ths?

53/64 is equal to 13.25/16 - so yes, if you rounded the number it would be 13/16

The fraction (in 64ths) times 16 or
(53 times 16) divided by 64

This is actually simpler, it's a FORMATTING question.

Click on the cell that has your 64ths showing
Press Ctrl+1 to open the Format Cell dialog box
On the Number tab, click Fraction for the category
Under Type select As sixteenths (8/16)
Click OK
All done! This method preserves the original data, too, not doing any math on it.