Alright, well, this clarify a lot of things, and changes a lot tools to add in your collection, also.
When you are dealing with audio, especially antique things you will need:
1. tube/valve tester
2. a decent soundcard + pc : it will became handy because when you actually fix things up, you can record thing up and monitor the levels. It will also give you a clear idea on what the problem may be. Also, if you are going to change the tubes and valves and stuff, you can see the difference.
3. a multi meter. A decent one. You are dealing with HV and you must not be cheap, otherwise you end up killing yourself.
4. a good/solid power supply: vintage always equals to more power consumption, and nasty voltages. Better be ready, since you don't know what kind of stuff you are going to test...
5. an oscilloscope. a basic oscilloscope would do most of your jobs. Take a combiscope 20Mhz, it's not that much and you won't regret it.No need of digital things.
6. an audio spectrum analyzer may be handy in some few situations (that's why I don't recommend it), but there's even something better for the price, see after...
7. mains filter. When you are working with the audio spectrum, low (0-200 Hz) lead to mains interference, menaing you will hear hums...
8. an audio signal generator. This may be very tricky to find. See after.
So, whenever you are dealing with HV, you MUST not be cheap on instruments, and you can't trust a 100 usd multimeter. Better considering a good one instead, security first.
As a reguard for power supply, anyway, take something at least 5A rated, I can't tell which voltages tought: never used valves, but 30/60V may be a good range... Also, take a mains isolated one.
As a reguard to mains filtering: yes, you will need some kind of filtering otherwise you may add the hum problems on your device under test (or restoration)
As for audio signal generators, you need something very specific, and even if the device is in specs it don't mean it sounds good... take something refurbished DC-250Khz... You won't need more, and it have to be very precise and low distortion.
As a reguard for the oscilloscope, if you don't like analogs and you want digitals take a basic one, don't spend too much since it's not worth it.
The most fitting solution to your needs instead of a spectrum analyzer or instruments of any kind exept your multimeter may well be an audio analyzer or a signal analyzer (better if specific for audio), like this units here:
http://www.ebay.com/itm/Panasonic-VP-7723A-5Hz-100KHz-0-0005-Audio-Analyzer-/380403386509http://www.datatron.it/user/articoli.php?op=det&id=624Don't forget that if you want to spend more, there's this one:
U8903A Audio Analyzer :
http://www.keysight.com/en/pcx-x205218/audio-analyzers?nid=-32511.0.00&cc=IT&lc=enand it's $14.000 (that's not much, considered that a decent spectrum analyzer is well over $20K), however, this is an example. You can find a lot of old stuff that it's more suited to the technology you are dealing with...