Yes, Sultan has winning strategies. Here are two examples:

First winning strategy. Sultan plays the following numbers in order: $2$, $3$, $4$, $6$, $8$, $20$, $24$ and he wins. Indeed, let $n \geq 100$ be Tarik's number and assume that Sultan will not win before playing the last number $24$.
- $n$ is not divisible by $2$, so it is congruent to one of the twelve numbers $1,3,5,7,9,11,13,15,17,19,21,23$, in modulo $24$.
- $n-2$ is not divisible by $3$, so it is congruent to one of the eight numbers $1,5,7,11,13,17,19,23$, in modulo $24$.
- $n-5=(n-2)-3$ is not divisible by $4$, so it is congruent to one of the four numbers $2,10,14,22$, in modulo $24$.
- $n-9=(n-5)-4$ is not divisible by $6$, so it is congruent to one of the two numbers $10,22$, in modulo $24$.
- $n-15=(n-9)-6$ is not divisible by $8$ and is congruent to $4$ in modulo $24$.
- $n-23=(n-15)-8$ is not divisible by $20$ and is congruent to $20$ in modulo $24$.
- Finally, Sultan ends up with $n-43 \equiv 0 \bmod 24$, which is still a positive number divisible by $24$, and Sultan wins.

Second winning strategy. Sultan plays the following numbers in order: $2$, $3$, $4$, $6$, $16$, $12$ and he wins. Indeed, let $n \geq 100$ be Tarik's number and assume that Sultan will not win before playing the last number $12$.
- $n$ is not divisible by $2$, so it is congruent to one of the six numbers $1,3,5,7,9,11$, in modulo $12$.
- $n-2$ is not divisible by $3$, so it is congruent to one of the four numbers $1,5,7,11$, in modulo $12$.
- $n-5=(n-2)-3$ is not divisible by $4$, so it is congruent to one of the two numbers $2,10$, in modulo $12$.
- $n-9=(n-5)-4$ is not divisible by $6$, so it is congruent to $10$ in modulo $12$.
- $n-15=(n-9)-6$ is not divisible by $16$ and is congruent to $4$ in modulo $12$.
- Finally, Sultan ends up with $n-31 \equiv 0 \bmod 12$, which is still a positive number divisible by $12$, and Sultan wins.