If you are willing to assume that $Y$ has a symmetric distribution within the two groups, then the medians of the two groups (i.e., $q_{21}$ and $q_{22}$) could be used in place of the means. Furthermore, if you are willing to assume that $Y$ is normally distributed within the two groups, then you could make use of the relationship between the IQR and the SD for the normal distribution, namely, $SD \approx IQR / 1.35$. So, you can compute the two IQRs with $IQR_1 = q_{31} - q_{11}$ and $IQR_2 = q_{32} - q_{12}$, transform them to SDs, pool those two SDs in the usual manner, and then you have all of the pieces to compute the standardized mean difference.
Example: For your example data, this would be $$IQR_1 = 174 - 58 = 116$$ $$IQR_2 = 158 - 31 = 127,$$ so $$SD_1 = 116 / 1.35 = 85.93$$ $$SD_2 = 127 / 1.35 = 94.07.$$ Therefore, $$SD_p = \sqrt{\frac{(80-1)85.93^2 + (46-1)94.07^2}{80+46-2}} = 88.97.$$ And finally: $$d = \frac{85-79}{88.97} = 0.07$$ Now you could use the usual equation to estimate the sampling variance of $d$ (Hedges & Olkin, 1985): $$v = \frac{1}{80} + \frac{1}{46} + \frac{0.07^2}{2(80+46)} = 0.034.$$
Remarks: Under normality, $d$ should be an okay estimator of the true SMD. However, the use of medians in place of means and the estimation of the SDs via the IQRs involves a loss of precision. The usual equation for the sampling variance of $d$ does not reflect that, so it yields values that are probably too small (on average).
Also, the appropriateness of this method hinges on the symmetry/normality assumption. Unfortunately, authors typically choose to report medians and IQRs whenever they suspect that $Y$ has a non-normal/symmetric distribution. So, I would regard this method only as a rough approximation.
References:
Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Orlando: Academic Press.
This at least gives some pointers for - and a partial answer to - this question.
In the case of sample quantiles, the standard error depends on which definition of sample quantiles you actually use. I believe R, for example, includes 9 different definitions of quantiles in its quantile function.
For cases where the sample quantile is an exact order statistic, the standard error of the sample quantile follows from the standard error of that order statistic.
If a quantile is based on some weighted average of two order statistics, then the standard error can be obtained from their variances, their covariance and the weights.
As a result, confidence intervals can be formed; in the case of a quantile being an order statistic, a binomial distribution can be used to form a nonparametric interval directly from order statistics.
Best Answer
The quantile or probit function, as you can see from the link (see "Computatuon"), is computed with inverse gaussian error function $erf^{-1}$ which I hope is downloadable for calculators like TI-89. Look here for instance.