The ignitor you can check using an (old) AM radio.
When the ballast with ignitor is powered without a lamp, the ignitor makes a lot of RF mess that could be picked up by the radio (mainly at the bottom end of the AM band). Prezence of a buzz signalizes the ignitor is trying to do something, but it still could have some HV insulation problem.
After you manage to pick up the resulting buzz, try to connect a piece of wire (0.5..1m spread across the desk; be careful to have its lose end well insulated) to the lamp live terminal. If the buzz becomes way stronger in the radio, it means the ignitor is generating the high voltage, so very likely is OK.
I also wouln't be surprized the lamp did not survive the transport after being used for some time (spot welds connecting the arctube tend to weaken over time and then crack way easier when the lamp is moved around). To test it you may also use one of the ijstant stsrt fluorescent ballasts. Normally these are able to ignite quite fatty discharge in HPS, so good to test the lamp integrity. Of course, you should not run it for longer than a few second, such low power is not able to warm up the electrodes so the HPSwill suffer on it, but for a brief check of the lamp it is good.