You cannot obtain a meaningful estimate of the standard deviation for a particular day if you only have one observation for that day. You do not observe any variation at all, so there is no way to tell what the standard deviation is. A GARCH model employs some restrictive assumptions (essentially, that is what the model is made of) to be able to estimate conditional standard deviations for each day. Ideally, you would like to compare that to standard deviations obtained from a model that does not use restrictive assumptions. But since you only have one observation a day, that is impossible.
You can still evaluate the statistical adequacy of your GARCH model by looking at the standardized residuals. They should have zero autocorrelation at all nonzero lags (Ljung-Box test may be used for that, though it becomes problematic if the conditional mean model is ARMA). The same applies to squared standardized residuals (Li-Mak test can be used for that; ARCH-LM is not suitable, however). Also, the probability integral transform (PIT) of the standardized residuals should be Uniform[0,1] (Kolmogorov-Smirnov test can be used for that). A statistically adequate GARCH model must pass all three tests. You could come up with additional tests, but these three are the standard ones.
For R code, see the
vignette and the
reference manual of the
rugarch package. Here are some examples by the author of the package.