Letters of UDAYPUR in alphabetical order: A, D, P, R, U, U, Y (U repeats twice).

Count words starting with each letter before U:

Starting with A, D, P, or R: remaining 6 letters with U repeating $\Rightarrow \frac{6!}{2!}=360$ each. Total $=4\times 360=1440$.

Now words starting with U (use first U; remaining letters A, D, P, R, U, Y — all distinct):

UA__: $5!=120$.

UDA__ with next letter before Y from $\{P,R,U\}$: UDAP, UDAR, UDAU each contribute $3!=6$. Total $=18$.

UDAY__: remaining letters P, R, U. Before UDAYPUR alphabetically: UDAYPRU $=1$. Then UDAYPUR is next.

Rank $=1440+120+18+1+1=1580$.