Understanding the physical process that takes place in this problem about ideal gases

We keep heating the cylinder, gas is coming out of it, and I have no idea what’s going on with the pressure inside the cylinder (it is certainly not decreasing, but does it stay constant during the process? If so, how do we know that?).

You confusion is well justified. IMO the wording of this problem is awful. Based on the book answer it appears that, during the leakage of the gas, the gas pressure is constant and equal to the valve opening pressure. In order for this to be so, the gas would need to leak out very slowly so that pressure gradients within the gas are minimized. That, in turn, means the temperature of the environment would have to increase very slowly to the maximum of 500 K. This, in my opinion, is the critical assumption made but not stated in the problem.

To sort it out, based on the book solution, the sequence appears to be as follows:

1. Initially, 4 moles of the gas are in equilibrium internally with a temperature of 280 K, pressure of 150 kPa and volume (computed based on ideal gas equation) of 62.1 m$^3$.

2. The temperature of the environment is increased to 500 K. The problem should have explicitly stated this. What's more, the temperature of the environment must be increased gradually so that the temperature of the gas and the temperature of the environment are in equilibrium. Under these conditions the internal equilibrium temperature and pressure now increase slowly until the temperature and pressure reach 373 K and 200 kPa, respectively, and the valve opens.

3. Although it is not stated, the gas must be leaking out very slowly, so that temperature and pressure gradients within the gas are minimized and the gas is in internal equilibrium. Again, if they stated that the environment temperature was slowly increased, then we can assume the leakage is slow. Otherwise, you would not be able to measure an equilibrium temperature of 500 K and an internal equilibrium pressure. Moreover, it is apparently assumed that the gas pressure during the leaking process remains constant at the opening pressure of 200 kPa.

4. When the gas temperature reaches equilibrium with the environment, 500 K, the valve is apparently closed, though nowhere is this stated in the problem. Alternatively, they want to know how much gas has leaked out up to that point. So the final pressure 200 kPa.

5. We can now apply the ideal gas equation to the initial and final states to determine the moles of gas remaining. Or

$$frac{n_{o}RT_{o}}{P_{o}V}=frac{n_{1}RT_{1}}{P_{1}V}$$

$$n_{1}=n_{o}frac {p_{1}T_{o}}{p_{o}T_{1}}$$

Which is your solution.

UPDATE:

would my solution be completely wrong in the case the environment had its own, fixed temperature?

I don't believe your answer would necessarily be wrong. And that's because when the gas eventually reaches an equilibrium temperature of 500 K, it should continue to leak until enough moles are removed so that given the remaining moles the pressure will be 200 kPa at a temperature of 500 K for the given fixed volume, per the ideal gas law.

My only concern for the rapid heating approach is the possibility that more gas will leak out than needed before the valve closes, since the valve may respond to local pressure as opposed to the equilibrium pressure. In other words, you could wind up with a final pressure less than 200 kPa and more than 1 mole of gas removed at 500 K. By slowly increasing the temperature the pressure can be made reasonably constant, allowing it to continually adjust itself by letting gas leaks out, and ending up at 200 kPa.

All of this, however, deals more with the practical considerations associated with the performance of relief valves, location of pressure sensing mechanisms, etc., none of which may be germane to an academic problem like this one. The exercise is clearly intended to be a straightforward application of the ideal gas law to initial and final equilibrium states. For that intention, the details of the process are probably unimportant.

Hope it helps.