You raise an interesting question. First thing first, if you have an observation of 0.35, a mean of 0.25, and a standard deviation of 1/10^7 (that's how I interpret your e^-7 bit) you really don't need to go into any hypothesis testing exercise. Your 0.35 observation is very different than the mean of 0.25 given that it will be several thousands standard deviation away from the mean and it will probably be several millions standard errors from the mean.
The difference between the Z-test and the t-test refers mainly to sample size. With samples smaller than 120, you should use the t-test to calculate p values. When sample sizes are greater than that, it does not make much difference if at all which one you use. It is fun to calculate it both ways regardless of sample size and observe how little difference there is between the two tests.
As far as calculating things yourself, you can calculate the t stat by dividing the difference between your observation and the mean and divide that by the standard error. The standard error is the standard deviation divided by the square root of the sample size. Now, you have your t stat. To calculate a p value I think there is no alternative than to look up your t value within a t test table. If you accept a simple Excel alternative TDIST(t stat value, DF, 1 or 2 for 1 or 2 tail p value) does the trick. To calculate a p value using Z, the Excel formula for a 1 tail test is: (1 - NORMSDIST (Z value). The Z value being the same as the t stat (or the number of standard error away from the mean).
Just as a caveat, those methods of hypothesis testing can get distorted by sample size. In other words, the larger your sample size the smaller your standard error, the higher your resulting Z value or t stat, the lower the p value, and the higher your statistical significance. As a short cut in this logic, large sample sizes will result in high statistical significance. But, high statistical significance in association with large sample size can be completely immaterial. In other words, statistically significant is a mathematical phrase. It does not necessarily mean significant (per Webster dictionary).
To get away from this large sample size trap, statisticians have moved on to Effect Size methods. The latter use as a unit of statistical distance between two observations the Standard Deviation instead of the Standard Error. With such a framework sample size will have no impact on your statistical significance. Using Effect Size will also tend to move you away from p values and towards Confidence Intervals which can be more meaningful in plain English.
If you are concerned about sample size and significance, good concepts to start out with include effect size and power (while on the topic of CIs, you might want to include accuracy as well)
As noted previously, 95% CI refers not to probability but to confidence; it is not the likelihood that the current CI contains the population parameter, but that out of 100 CIs 95 CIs will succeed in capturing the population parameter. The true probability remains unknown (unless you take a Bayesian approach), see Explorations in statistics: confidence intervals.
Best Answer
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):
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.)