Absinthe is one of the purest forms of alcohol ever consumed, and very hard on the mouth if you drink it! The sugar is to 'soften the blow'!
Just put the cube onto a teaspoon and dip into absinthe quickly before gulping.
(or apply a few drops from a bottle onto the cube). I've heard you can also set light to the sugar cube so that some of the excess alcohol fumes burn off like a christmas pudding effect, but I'm not sure that this was really common practise.
(Then it's more like a peppermint! but with a sting in the tail)
It is banned in many countries because the effects are almost 'hallucinogenic' and the after-effects like a Class A narcotic, also known as 'visiting the green fairy'.
Toulouse Lautrec and co. were crazy about the stuff, probably crazy because of the stuff too!
(see Kylie Minogue in 'Moulin Rouge').
Not to be attempted by the faint-hearted! (even if it IS legal in your area!) ;-)
I still think that it would be useful to know if there is any lip shaped into the neck of the decanter. Its size may give a clue, since a perfume/cologne decanter should typically produce quite a thin stream of fluid.