This is the "midcentury" version Santa Cruz Rum Fix as described by David Wondrich in his book Imbibe!. Midcentury refers to the mid-19th Century when Jerry Thomas published his first edition of The Bartender's Guide in 1862 which also contains this drink recipe.
There is a small addition made to the Rum Fix in the 1887 edition of Jerry Thomas's book - 1 tsp of Orange Curacao. For me, this improves the drink slightly. If adding the Curacao, cut the sugar back to a scant teaspoon to retain balance.
As with all Fixes, your base spirit needs to be a quality product. There is a lot of base spirit compared to other ingredients so it will shine through (or ruin the drink). Back during Jerry Thomas's day they used Santa Cruz rum in this drink. Though aged, Santa Cruz rum was the lighter of the two styles of rum most often mentioned - the other being Jamaican rum. So you can use a nice Virgin Islands aged rum to be authentic, but for this drink it's really up to you and what you like. If you like funk-bombs by all means use the pot still Jamaican.