The simple answer is you measure the orbit, then the orbit gives you the perihelion/aphelion. James K said this in a longer way; you had already postulated this, and found it unsatisfying.
If you want to get pedantic, you blast the object with radar (possibly lidar in some future time), directly measure range (from time delay and speed of light), and then try to get range rate from Doppler shift (change in frequency of return signal). Given a crude orbit (a ballpark r value, or distance from Sun- not hard if you don’t need decimal places), and the known state of the Earth (in its orbit relative to the target, at the time of the observation), the value for range rate will likely give ‘increasing r’ (not yet at aphelion) or ‘decreasing r’ (pre-perihelion). Of course, flatlining r per se tells you nothing (at that time).
In practice, planetary radar only worked out to Saturn due to design limits, and that was Arecibo (RIP). Inside of Saturn, planets are known fairly well, I’d say. Small bodies are points of light in all but the most humongous (optical) telescopes, so James K’s process works fine; we simply spend two or more lunations to establish an overall orbit (depending on how many decimal places are necessary). Few things compel us to hurry any more than this; the exception is a small body that, after a first, crude orbit solution, may be Earth-crossing, and somewhat close at that.
In that case, we would resort to precovery- searching archived images for the object, to extend the observing arc, and put more decimal places on the orbit solution. If even that still gives a close pass by Earth, then the schedule of a planetary radar might be overruled to observe the target. Combining both the optical solution (precise in 2D, plane-of-sky) and the radar values (lousy in 2D, but precise in that third D) will give plenty of decimal places on the next orbit solution, and thus the apsides. Of course, these small bodies are not planets (the term “minor planet” is now deprecated) but should some hollywood planet magically appear, that’s what we’d do.