How do I calculate how many newtons a hollow rod can carry without buckling?

For Euler Buckling the critical load is:

$$ P_{cr} = left(frac{pi}{K L}right)^2 E I $$ where:

• $P_{cr}$ is the critical load
• $E$ is Young's modulus (for steel assume 200 [GPa])
• $I$ is the second moment of area. ($frac{pi}{64}(d_o^4- d_i^4)$
• $d_o, d_i$ is the outer and inner diameter respectively
• $L$ is the length of the rod
• $K$: is a parameter that depends of how each end of the beam is constrained.

K is the most difficult to determine. For your case, would be 1.

Additionally, regarding the question of the yield stress $sigma_y$. The way you use it, is that you need to check whether, the $P_{cr}$ is greater than the $sigma_ycdot A=sigma_ycdot frac{pi}{4}cdot(d_o^2 - d_i^2) $. If it is greater then yield occurs earlier than buckling.