Note this answer only works if reaching 1000 does not warrent a win. This was the original question but since it was changed to "greater than or equal to 1000" is a win it is incorrect.
Lets look at this for any $N$
if $N>250$ player one can multiply by $4$ and win instantly so label all of those $F$.
If $251>N>82$ it will depend on if $N$ is even or odd. Both players will just add one to prevent the other getting a $250+$. If $N$ is even it is $S$ while if it is odd it is $F$.
If $83>N>63$ then if $N$ is even it is $F$ while if it is odd it is $S$. The reversal happens because when $N=82$ then $3N=246$ which is less than $250$ but even so by the previous paragraph the player awarded that will lose. Simular strategys arrise by multiplying any $N$ by $3$.
If $63>N>20$ then the first player always wins because he can multiply by $4$ to yield a $N>83$ which is even which means he wins.
If $21>N>7$ it is alternating again as if $N$ is even it is $S$ while if it is odd it is $F$. This is because multiplying makes the other player win so they can only add.
For all $N$ between $1$ and $10$ besides $1,8,10$, the first player can force the number into a region which is an $S$ by making sure it is even and between $20$ and $7$. If he cannot, the second player can force a win. This means if $N=1,8,10$ then the second player wins. If $N=2,3,4,5,6,7,9$ then the first player wins.