How to automatically sum up values within buffers?

Use the Spatial Join tool in the Analysis toolbox->Overlay toolset with these settings to generate statistics for all of your concentric buffers at once.

1. Target Features: Concentric Ring buffer feature class

2. Join Features: Population points

3. Output feature class: Specify the name and location you want

4. Join Operation: Join_One_To_One

5. Join Type: Keep All

6. Field Mapping - In the actual tool dialog right click the Population field and change the Join Type to Sum. You can remove the other point feature class fields or keep them and set up other summaries as well. Set this up in a model using the tool interface and then export that to a python script to see the syntax of this parameter for Python.

7. Match Option: Intersect

8. Search Radius: 0

9. Distance field: Not applicable so do not include it.

This will output a duplicate of your concentric ring buffer feature class with an added Population field containing the sum of all the population points it intersected.