Note: We have a separate question on whether this is safe, Is copper safe for long-term food storage?.
Bust out your Moscow Mule mug and keep your syrup in that. Mold is exquisitely sensitive to copper. The amount that leaches into the syrup will prevent mold growth, I here assert.
I am back, with some science. Especially for you, @Esther because you doubted and you were right to doubt. This premise is testable experimentally and that is what I did.
Here is the setup as of 11/2 - 1 week ago. I made the sugar syrup with a 1:1 sugar and water mix, heated in a saucepan. Jars were rinsed first. Approximately 150 ml syrup in each of 8 jars. I did not have a copper cup but I had a silver one - also 150 ml.
To simulate a copper metal cup I used pennies.
These are American pennies which I washed with dishsoap and rinsed. First 2 jars (0) were controls with only syrup, no pennies. Subsequent jars had 2 pennies (x2 jars), then 5, 10, 20 and 30 pennies.
Experiment is ongoing. Here are results as of this AM 11/9 at 1 week.
Jars with only syrup and no pennies had mold colonies by day 3 and they are easy to see now. The mold looks like aspergillus (bread mold) and is floating on top.
Jars with 2 pennies had a few mold colonies.
Jar with 5 pennies developed mold for the first time this morning. I put red arrows on the mold colonies.
Jars with 10,20 and 30 pennies have no mold colonies yet. Neither does the silver cup. below: 10 and 20 penny cups (30 pennies not shown) and silver cup.
For sugar syrup left open at room temperature. addition of metal as copper or silver retarded mold growth over the experimental period of 1 week.
You could do this now as pennies. Or maybe a piece of scrap copper pipe from your shop? A classier way would be to get a little silver creamer pitcher which would both retard mold as my silver cup did and be easy to pour from into your coffee.
The most vexing thing now is how to get people to revisit this question (now invisible from the downvotes) and replace those heartrending downvotes with sweet smelling upvotes!