To explain the question: I ran 500 meters uphill. This costs energy, which can be measured in Joules etc.
My question is: how far should I run on flat ground to expend the same amount of energy?
Approximately 750 m.
Any physical activity has a MET value associated to it. The definition is not important; what matters for us is the following:
(...) MET can be thought of as an index of the intensity of activities: for example, an activity with a MET value of 2, such as walking at a slow pace (e.g., 3 km/h) would require twice the energy that an average person consumes at rest (e.g., sitting quietly).
A fortunate coincidence is that running on flat ground gives MET values that roughly coincide with the speed in km/h. For example, running at 12 km/h has a MET value of approximately 12 (it's actually 12.5). To compare that with running uphill, we need a MET table with values for different inclines. One is given by American College of Sports Medicine: Guidelines for Graded Exercise Testing and Exercise Prescription, 2nd Edition, Lea & Febiger, Philadelphia (1980) and reprinted in this article (the grades for outdoors are wrong, they're all halved). I found a formula that reproduces all those values with less than 0.7% error:
MET = 1 + (0.957*v - 0.064)*(1 + 0.045*g)
where v is the speed in km/h and g is the grade in percentage (e.g., for 7% incline, g = 7).
We really don't need to be too precise because there are so many other factors involved, so there's no problem simplifying the formula above to, say:
MET = v*(1 + 0.05*g)
That is, add 5% to the calories burned when running for every 1% of incline. In your case, it's 10% incline, so +50% calories. That's the same as running +50% of the distance: 500 + (50% of 500) = 750.
You could also try an online calculator (but some of them are way off). I tried this one with some typical values and the OP's weight:
Sounds about right.