Simple Syrup
The original recipe calls for one sugar cube, being muddled with some water. This method should allow the cocktail to evolve over time, getting sweeter.
Well, kind of. The sugar will not dissolve completely and you end up with granulated sugar in your drink, which is preferred by some people.
I like to have a bit more control over the experience. The drink has to taste the same, from the first to the last sip. Therefore, I use Simple Syrup. And if you do other cocktails as well, you have Simple Syrup around anyway.
Also, I have a sweet tooth and like to use a rich (2:1) Demerara Simple Syrup, which offers a more complex flavor profile. Nevertheless, feel free to try any Simple Syrup that suits you best. If its a 1:1, 1:1.5 or 2:1 ratio with "normal" sugar, Demerara sugar or maybe even Agave Syrup combined with a Tequila.
The Spirit
Actually, most Bourbons or Ryes will do the job. If it's your Bulleit 95 Rye Rittenhouse Rye, Marker's Mark, Wild Turkey or Buffalo Trace, to name some of the generally most available. High proof Bourbons or Ryes will stand up to the other ingredients and work great too.
In general use the recipe as a base for your own variation. Play around with the ingredients and ratios. If you're in the mood for a smoky/peaty Old Fashioned, more power to you.
The Ice
You could use any ice you have around, but actually, you don't want to. You want your drink to be chilled and diluted, but not over-diluted, which waters down your drink too much, reducing the taste. Small ice cubes dilute to quickly. Use a big ice cube or big ice ball for keeping the dilution rate at a minimum.
For example: If you use a Tumbler with a diameter of 2.76inch/7cm, an ice ball, preferably clear ice, with the diameter of 2.36inch/6cm is a good fit for chilling, minimal dilution and great looks.
The Finish
You might skip the Luxardo Cherry, maybe even the Orange Peel, but don't skip expressing the oils of the Orange Peel. This elevates the cocktail to a new level. It balances the cocktail out and brighten it up without any negative effect on the other ingredients.