No. In a move, if we choose $100$ cups where $k$ of them are facing downward and $100-k$ of them are facing upward, then the number of cups facing downward is changed by $(100-k) - k = 100 - 2k$. Since there is an even number of cups facing downward initially, there is always an even number of cups facing downward. Thus, it is impossible to make all $1999$ cups face downward.

It is possible to make all cups face downward if there are $1998$ cups. If we turn over cups $1, 2, \ldots, 100$ and then turn over cups $2, 3, \ldots, 101$, we see that only cups $1$ and $101$ are turned over. This shows we can turn over any $2$ cups after $2$ moves. Since $2 \mid 1998$, we can repeat the same process to turn over all cups.