Not just for the Derby. This source, which explains why we can thank the Arabs for the basis of this refreshing adult beverage along with inventing Algebra, notes:
Mint Julep first appeared in print in 1803 described as a "dram of spirituous liquor that has mint in it, taken by Virginians in the morning."
A breakfast tonic to tone ye olde brain. Healthy herbs. You can use either homemade sugar syrup or confectioner's sugar. Here's one. Use a chilled julep cup or collins glass. I grow a mint called "Kentucky Mint" which is quite intense, but close to Spearmint, just for this purpose, but I haven't made a batch in years. I should. Unlike Southern families, I do not own any heirloom Julep Cups. I am tempted by these on eBay ($500 for a pair). Nice Sterling cups seem to run between $100 to $1000 apiece. I am not in the market for these right now.