Simple interpretation of effective population size?
Like, would it be overly simplistic to say that the sampled individual could be expected to contain ~1/1000 of the population's standing variation?


It is a little unclear what you mean by "variation" here. If you mean "genetic diversity" aka "expected heterozygosity", then it would not just be simplistic, it would be wrong to say that!


  Or to say that a given (observed) nucleotide state has a 0.001 probability of being fixed?


Ah that one is pretty good. If the effective population size $N_e = 1000$, then the probability of fixation a a new neutral mutation is $1/2N_e$ (assuming diploidy). The probability you indicate would be correct in a haploid population if you are talking about a new neutral mutation.

Another simple statistic you might want to consider is that, in absence of mutations, the genetic diversity would decrease by a factor $\frac{2N_e - 1}{2N_e}$ at every generation (assuming diploidy again).

You can derive a whole bunch of statistics of interests. For example, the expected genetic diversity in the total population at a locus at which the mutation rate is $\mu$ is $\frac{4N_e\mu + 1 }{4N_e\mu}$. You can derive many other statistics you might be interested in such as the expected number of segregating sites in a sample of size $k$ or the expected coalescent times for examples.

For more information just have a look at an intro course to population genetics. You'll find book recommendation in this post.