Spider hunting flies 
A spiderweb is a square with $100 \times 100$ nodes. $100$ flies caught into the web stacked at $100$ different nodes. A spider which was originally at the corner of the web crawls from a node to an adjacent node counting moves and eating flies on its way. Can the spider eat all flies in no more than $2100$ moves? or $2000$ moves?

Thanks in advance.
 A: As has been pointed out, your question is very unclear. In my answer I shall assume that you start at the corner $O$ of an $n \times n$ square grid of nodes and you can only move along edges between (orthogonally) adjacent nodes. Then the maximum number of moves you need to reach any given set of $n$ target nodes in the grid is at least $(r^3+2s-r)$ and at most $(2ns-s-1)$, where $r = \lfloor\sqrt{n}\rfloor$ and $s = \lceil\sqrt{n}\,\rceil$. Note that both bounds are $Θ(n \sqrt{n})$ as $n \to \infty$.
Firstly, if the target nodes include an $r \times r$ subarray of nodes with spacing of $r$ nodes between each row or column, positioned as far from $O$ as possible (*), then you need at least $r$ moves to get from one node in the subarray to another, and at least $2s$ moves to get to the first node in the subarray, and hence you need a total of at least $(r^2-1)r+2s = r^3+2s-r$ moves.
Secondly, we can always divide the grid into $s$ rectangular subgrids in a vertical stack (each having width $n$ and height at most $s$). We can move through the subgrids one at a time, from the closest to the furthest from $O$, and in each subgrid we visit the target nodes in that subgrid in order of their horizontal position, starting and ending at nodes on the left/right boundaries, alternating between going from left to right and going from right to left each time we go to the next subgrid. We can complete each subgrid using at most $(n-1)$ horizontal moves and $k(s-1)$ vertical moves, where $k$ is the number of target nodes in the subgrid. We can traverse between subgrids using at most $(n-1)$ vertical moves. Therefore we need at most $(s+1)(n-1)+n(s-1) = 2ns-s-1$ moves.
(*) In case it is not clear enough, we are considering the case when the target nodes include $r^2$ nodes placed in a square configuration where adjacent target nodes are $r$ moves away. Of course we want to place this configuration as far from $O$ as possible. One can get a slightly better lower bound by adding in the $(n-r^2)$ leftover nodes somewhere, but I am too busy to work out the details, and they don't matter much.
A: Utilizing @Alexis Olson's mathematica code, I came up with the following distribution for path lengths (shown in red) and plotted it against the normal distribution (shown in blue, using the mean & standard deviation of the path lengths). Seems to agree remarkably well (tested on 5000 paths). 

A: I had a different solution from the answer by user21820 for the asymptotic upper bound for an $n\times n$ grid. I'll just briefly sketch it here:
Take the left-most remaining $\approx\sqrt{n}$ points (which we will call a block), and traverse them from top to bottom, going left/right where necessary. Repeat, but switch vertical direction alternately. Total vertical movement is $\text{number of repeats}\times (n-1)$ and total horizontal movement is $\le\sum_{\text{block}}{\text{width of block}\times(\text{number of points in block}+1)}$. Maximum is $\approx 2n\sqrt{n}$.
Edit: This solution is slightly worse than the other one, and hence cannot answer the question about 2000 moves.
