The outcome variable that you want to predict here, number of drinks consumed per month, is a count variable. The fact that many potential predictors are binned is inconsequential; it is the distribution of the outcome that matters for determining what kind of regression modeling is needed. Here, since the outcome is a count, you can consider regression methods that are appropriate for count variables, such as Poisson and negative binomial regression.
In my experience, Poisson regression is not a good choice because real count data like these are almost always over-dispersed, meaning that they exhibit more variation than can be accounted for by a Poisson model. So you might start with Poisson regression but move to negative binomial after. The model fits can be compared via likelihood ratio test for nested models or with fit indices such as AIC and BIC.
When considering counts of events that are rather rare you may have an overabundance of zeroes. This might happen if with past 30 day alcohol but would be more likely with something like past 30 day steroid use (or some other, more rare drug). In those cases, zero-inflated or hurdle models can be useful, especially if you have other covariates in the model.
Here is a good resource.