Like most questions, this is an "ask your spiritual father" one.
As has been noted, Slavic practice generally allows for beer on "no wine" days. In Greek practice, "no wine" is usually interpreted as "no alcohol". The difference in Greek and Slavic practices with regard to beer consumption on "no wine" days may be largely related to how different cultures and regions related to beer and wine historically. Others have mentioned how beer became somewhat of a staple in Western Europe and other areas where water contamination was a major concern. In such contexts, beer consumption was a matter of sustenance and survival, so allowing beer in such contexts during "no wine" days is easy to understand. It should also be remembered that the beer consumed for sustenance in these contexts may also have been of fairly low alcohol content compared with commercial beers that are typically consumed today.
It seems that beer never had much of a prominent role in Greek society as it did in other parts of the world, and this may at least partly explain the “no wine = no alcohol” rule in Greek practice. I have a better understanding of the role of beer in Western Europe historically than I do of its role in Slavic history. It would be interesting to know something about the beer that was allowed in Slavic practice on “no wine” days, and what the role of beer was in Slavic culture at the time.
At one time, Russian monasteries were famous for their meads. I assume that mead was considered a “honey wine” and thus not permitted on “no wine” days, but this would be interesting to know for sure. It would also be interesting to know if the allowance of beer on “no wine” days extended even to the Russian Imperial Stout styles imported from England, which could be around 10% alcohol.
Personally, I think that the “no wine = no beer” rule is a good one, especially keeping in mind how modern society relates to beer. Probably all of us on this list have access to a good source of water and plenty of food, so beer consumption for us is probably more of a matter of pleasure and luxury rather than sustenance and survival. The practice I have been given to follow is that “no wine = no alcohol”, and that I should also abstain from alcohol the day before receiving communion. I don’t say this to imply that anyone else should do the same (go ask your own spiritual father what you should do!), but I do personally find this rule to be very helpful in observing the fasts and in preparing for receiving communion. As a home brewer, I do appreciate a good beer and am never without large quantities of very excellent beer, as well as hard ciders and such. Yet, as one who also wants to make sure that I am ultimately dependent only on God, I appreciate these days of required abstinence as an excuse to exercise discipline and to ensure that alcohol does not have a greater role in my life than it should.