More context might change my interpretation, but here's how I (native AmE) read it.
At first, I thought run was meant in the sense of flee: run away. Maybe Harry had stolen the money. Criminals are said to "run from the police" if they're trying not to get caught.
When I got to the third sentence, it started to seem that Harry didn't have anything to fear. Harry's problem was that he had just gotten a lot of money, so he was very happy and excited and it was hard for him to decide what to spend it on first. People speak of "running to the store" to mean making a quick trip to a store to buy something. In context, I understand run to suggest not only going somewhere to buy something, but also "move about in a hurried and hectic way"—with glee. It's not unusual for a word to be chosen to suggest more than one of its senses simultaneously.
The word where indicates location, but it's not clear if the location would be a destination that Harry would run to, or a location that Harry would run in. The word "run" has many senses, and further context could suggest yet another sense beyond the primary sense of moving fast on foot. Here is an imaginary context that could suggest a different sense. Suppose that Harry wanted to run for office as a politician (that is, try to become elected) but he didn't have enough money—until now. Then "where" could refer to the district where he would run for office. That's unlikely, though. In the absence of some additional information like that, I think your guess is the best—plus, maybe, a suggestion of deciding where to spend the money first.
Other examples
Here are some other examples where people have used the phrase "where to run first". These might help clarify how context shapes its meaning. This kind of thing is common in English, especially with common "little words" like "run".
When I walked into Bourbon Battle, I couldn’t decide where to run first. On the right, guests were ushered to a blind tasting table where we sampled shots of fine bourbon and put a token in the jar of our favorite. On the left, Food Network’s Rossi Morreale cheered on mixologist contestants as they stirred and shook their versions of classic bourbon cocktails.
Flanking each side of the bartending stage were banquet tables of upscale southern cuisine. The menu included braised short rib quesadillas, coffee-BBQ spare ribs with sauerkraut, honey-drizzled fried chicken biscuits—you get the picture. I all but sprinted to the comfort-food section. Once I had procured four quesadillas and a splash of bourbon, I started to explore.
Source: The blog Hungry Lobbyist.
The author has just entered a large room where people are displaying various bourbons, bourbon cocktails, and foods. She is filled with glee and she has so many appealing alternatives to choose from, it's hard for her to choose which one to approach first. The feeling of glee makes her feel like running—but she's not really going to run. In the next paragraph, she says that she all but sprinted to one section. That's a figure of speech. It means that she didn't really run, but she was so thrilled, it was hard to contain her excitement as she walked. Really, "I couldn't decide" is a figure of speech, too. Obviously, she did decide.
Upon entering, there is competition between the view and the wafting aroma of the chefs’ tables as to where to run first.
Source: The blog Neighborhood Seen.
The author is describing a room with many appealing things in it. "Running" suggests that you would be happy and excited and would want to run gleefully to one of the alternatives that "compete" to be your first destination. Really, of course, you probably wouldn't literally run.
Upon arriving at Woodloch, our children did not know where to run first. Was it to the Bumper Cars, or the Indoor Forest or Bingo or to play Family Feud?
Source: The Moms.
The children are so filled with glee when they enter an amusement park, they want to run to a ride (I think), but there are so many good options, it's hard to choose which one to do first. When applied to children, "where to run first" in this gleeful sense probably makes sense literally. Applied to adults, it figuratively suggests the gleefulness of children in a situation like this.
Orb weavers [a kind of spider] usually sit at the center of the vertical web, with their head facing down. They can feel the vibrations throughout the signal line, so if an insect lands in a far corner of the web, the spider knows right where to run first.
Source: Woodland Park Zoo Blog.
This time, there are many alternative directions for the spider to run, most of which don't lead to the prey, but the vibrations tell the spider the exact direction to run to get it.
Vinny is a 5-year-old Frenchie [a kind of dog]. About 3 months ago he was napping and suddenly woke up startled, scared, yelping, couldn’t figure out where to run first and emitting the oil gland smell.
Source: A question posted to D.E.L.T.A. Rescue.
This means that the dog is frightened and is looking around for someplace to run for safety—to hide.
