To solve this question, we can apply divisibility tests. Start by checking 1261 for divisibility by prime numbers.

First, 2, 3, and 5 can be easily ruled out as divisors. Next, we check divisibility by 13. Dividing 1261 by 13 gives 97, which is a prime number.

Thus, the prime factorization of 1261 is: $1261 = 13 \times 97$

Now, observe that 91 is divisible by 13. Additionally, 97 is already a factor of 1261, meaning that 1261 will divide the expression $(91 \times 92 \times 93 \times 94 \times 95 \times 96 \times 97 \times 98 \times 99)$ completely.

Therefore, the remainder when this product is divided by 1261 is 0.