Yes, it can be done, if you use Fisher's R-to-z transformation. Other methods (e.g. bootstrap) can have some advantages but require the original data. In R (r is the sample correlation coefficient, n is the number of observations):
z <- 0.5 * log((1+r)/(1-r))
zse <- 1/sqrt(n-3)
min(pnorm(z, sd=zse), pnorm(z, lower.tail=F, sd=zse))*2
See also this post on my blog.
That said, whether it is .01 or .001 doesn't matter that much. As you said, this is mostly a function of sample size and you already know that the sample size is large. The logical conclusion is that you probably don't even need a test at all (especially not a test of the so-called ‘nil’ hypothesis that the correlation is 0). With N = 878, you can be quite confident in the precision of the estimate and focus on interpreting it directly (i.e. is .75 large in your field?).
Formally however, when you do a statistical test in the Neyman-Pearson framework, you need to specify the error level in advance. So, if the results of the test really matter and the study was planned with .01 as the threshold, it only makes sense to report p < .01 and you should not opportunistically make it p < .001 based on the obtained p value. This type of undisclosed flexibility is even one of the main reasons behind criticism of little stars and more generally of the way null-hypothesis significance testing is practiced in social science.
See also Meehl, P.E. (1978). Theoretical risks and tabular asterisks: Sir Karl, Sir Ronald, and the slow progress of soft psychology. Journal of Consulting and Clinical Psychology, 46 (4), 806-834. (The title contains a reference to these “stars” but the content is a much broader discussion of the role of significance testing.)
You are probably looking for the delta method, see for example:
Oehlert, G. W. 1992. A note on the delta method. American Statistician 46: 27–29.
Best Answer
You can do it with $n$ Monte Carlo permutations: you permute the values in $X$ and in $Y$ to break their relationship and compute MIC, $n$ times. Then, you can then compute the $p$-value as: $$ \frac{1}{n}\sum_{i=1}^n I(\mbox{MIC}(X_{p_i},Y_{p_i}) > \mbox{MIC}(X,Y)) $$ where $I$ is the indicator function, and e.g. $X_{p_i}$ is the $i$th permuted version of $X$.
In practice you are counting how many times you can obtain MIC higher than your particular value just with a random relationship between $X$ and $Y$.
If you really want to use the tables at the link you propose, you have to download the table for your sample size (that is 30) and look up your MIC value 0.1643. Given that the last line in the table is
it means that your value would obtain $p$-value less than 0.05.