For any 2 distinct points $A, B$, if there are less than 4 points between them (from either clockwise or anticlockwise), it is easy to see that it is impossible find 4 other segments, which, along with $AB$ are pairwise intersecting. Also, for any 2 distinct points $A, B$ which have at least 4 points between them (both clockwise and anticlockwise), you can always find 4 other segments which are pairwise intersecting. Hence, if we label the points $A_1, A_2, \cdots A_{1000}$ in order, then $A_i, A_j$ can't have the same color if there are less than 4 points between them in either direction, and further, $A_i, A_j$ can always have the same color if there are at least 4 points between them in both directions, so we may colour $A_i, A_{i+1}, A_{i+2}, A_{i+3}, A_{i+4}$ the same color, and no other point may have the same color. So there are at least $\frac{1000}{5} = 200$ colors used. To see if 200 colors is possible, consider the coloring where all points $A_k$ with $\lceil \frac{k}{5} \rceil$ are equal are colored with the same color.

@above Am I getting the question wrong or something, or you are wrong? We claim that $k=143$ is the minimum value that $k$ can take. Indeed, if $k\leq 142$, then by pigeonhole there exist a color which is used at least $8$ times. WLOG let those points be $A_1 , A_2 , \dots , A_8$ in clockwise order and let there be at least $2$ points between $A_1$ and $A_8$. But then taking the segments $A_1A_6$ , $A_2A_7$ , $A_3A_8$ , $A_4A_i$ , $A_5A_j$ (Where $A_i , A_j$ are between $A_1$ and $A_8$ and $A_i$ is closer to $A_8$ then $A_j$) gives us a contradiction because there are only $2$ segments each of which has its endpoints colored in different colors. To see that $k=143$ works you can see the attachment. (Colors from $1$ upto $142$ are used $7$ times and color $143$ is used $6$ times.)

Attachments:

hakN wrote: @above Am I getting the question wrong or something, or you are wrong? We claim that $k=143$ is the minimum value that $k$ can take. Indeed, if $k\leq 142$, then by pigeonhole there exist a color which is used at least $8$ times. WLOG let those points be $A_1 , A_2 , \dots , A_8$ in clockwise order and let there be at least $2$ points between $A_1$ and $A_8$. But then taking the segments $A_1A_6$ , $A_2A_7$ , $A_3A_8$ , $A_4A_i$ , $A_5A_j$ (Where $A_i , A_j$ are between $A_1$ and $A_8$ and $A_i$ is closer to $A_8$ then $A_j$) gives us a contradiction because there are only $2$ segments each of which has its endpoints colored in different colors. To see that $k=143$ works you can see the attachment. (Colors from $1$ upto $142$ are used $7$ times and color $143$ is used $6$ times.) Yes I am wrong, sorry. I thought that for any 5 intersecting segments, every segment must have different colored endpoints.