Whoops, if you are not that familiar with the second quantization, we of course have to solve the problem without it first :) Earlier you asked about using the Pauli paramagnetism formula, and yes, you are correct, the formula is applicable. But you should really try to understand the derivation: check any classical text (Ashcroft-Mermin/your textbook) on Pauli paramagnetism and make sure that you understand the derivation of the susceptibility. You will notice that the derivation is normally made for a free electron gas in a box (\epsilon_p=p^2/2m), but that the final result only makes use of the density of states at the Fermi level. This is because the change in the spin-up and spin-down electron densities is directly given by the density of states times the Zeeman splitting. So in your case, the main problem is to calculate the density of states for the tight-binding model.
And yes, the zero-field susceptibility is obtained by setting h=0 in the end. But if you check the derivation, you will notice that the final result for the susceptibility seems to be independent of h! Can you see why that happens?