More specifically, you'd probably want an audio analyzer for such a task, but it likely doesn't have to be one of the more expensive dedicated units. Basically these are FFT analyzers that focus on low frequencies and analyzer their full bandwidth continuously, whereas swept spectrum analyzers cover a broader range and selectively catch the frequency area they are measuring with each point on the curve. An audio analyzer will also be optimized for low noise performance, typically offering more dynamic range at lower frequencies and offering audio friendly terminations, connectors, and drive levels. An audio analyzer also includes at least one source, so that you can effectively perform network analysis on your DUT, which is probably what would be most beneficial in your designs. Being able to plot the frequency and phase response of the device under test should tell you a lot about the unit you're working on. The 3582A is a dynamic signal analyzer, which is basically the same idea as an audio analyzer in terms of operation but without the integrated source.
This all said, you can get good performance doing all of this using a conventional PC and sound card with a couple of caveats. First, you'd need to select software suitable to the measurements you were making, there are some choices available. Second, there is no real input protection and not much level ranging on a sound card, so the signal you'd be measuring would have to be at the right general level to get good measurements and if you were just probing around in a tube circuit (with high voltage present) there would be a risk of damaging your sound card.
Another option would be a high resolution scope and signal generator, something with 10 or 12 bits of ADC input would be high enough detail to show you most of what could be heard, especially if using high resolution mode, and it would include the input protection and ranging that is useful for general measurements. Any modern scope also includes an FFT mode and some include a signal generator, the two of which can be used to measure frequency and phase though not always on the unit itself. Standard 8 bit scopes will still work fine, but there will be fine detail in the signal that you can definitely hear that doesn't show in the limited dynamic range of the converter, even with high resolution mode. There are some other things in this general category which could work the same, like the often recommended analog discovery, but the reality is you have quite a few options, so unless you need very high resolution or specific features, you likely already have something that can do the job in some configuration.