How to compute core stress in honeycomb structure?

This is just "stress = force / area" where the area is the cross section area of the honeycomb walls.

The data sheet for the honeycomb will give the wall thickness. To understand the data sheet, you need to know how honeycomb is manufactured: flat strips of material are bonded together in the appropriate places, and the pack of strips is them pulled apart to create the hexagonal spaces.

So the wall thickness in one of the three directions along the sides of the cells is double that in the other two directions, because that is where the strips were bonded together.

If all the cell walls were thickness $t$, each individual cell would be surrounded by a wall of thickness $t/2$ (since each wall is the boundary of two cells), but in fact two parallel walls of each cell have thickness $2t$ not $t$ so the "average" wall thickness surrounding an individual cell is $2t/3$ not $t/2$.