sarsparilla could be called a "root beer" - though what in the US is now known as root beer is distinct from sasparilla. (Sasparilla flavouring is an ingredient of the classic root beer - but in "sasparilla" it is the primary (possibly only) flavour)

the 'with a dash of cherry' means that the drinks were being mixed to order - as many fountain drinks were even up through the 60's; the syrups could be mixed and combined, or a shot of syrup added to a drink you'd already mixed.