Bomb Calorimeter vs Cup Calorimeter
Even though the pressure remains constant, heat can get in (or out) of the system. From the cup calorimeter, we are trying to find $\Delta H$ which is equal to the heat evolved from the system at constant pressure. We measure this heat and our job is done.

$$\Delta H=q_p$$

If we want to calculate $\Delta U$, we just to find the heat evolved from the bomb calorimeter, which can be done by measuring the change in calorimeter's final and initial temperature given that the heat capacity of the calorimeter is known.

$$\Delta U=q_v=nS\Delta T$$