Are you good at Summation?

We will have $\lceil \log_{\sqrt{2}} k \rceil - \lfloor \log_{\sqrt{2}} k \rfloor = 1$ for all positive integers $k$ except for the non-negative integer powers of $2,$ for which the difference is $0.$

Thus $N = \displaystyle\sum_{k=1}^{1000} k - \sum_{n=0}^{9} 2^{n} = 500*1001 - 1023 \equiv \boxed{477} \pmod{1000}.$