Absolutely do the compression test. And, I'd squirt some oil (engine oil, layup fogging oil, marvel mystery oil, anything) into ALL cylinders and crank it through it help make sure any moisture is pulled out and the rings don't start to rust up and stick from that water. That can happen very quickly if you let it sit with water in them.
If the compression test is good...it just means too much cranking pulled in water from the water lift exhaust. Memo: If the engine doesn't start right away, CUT OFF THE COOLING WATER before doing any more cranking. Then open it up after the engine starts.
Starters can and will be damaged by as little as 30 seconds of cranking, typically they can take 15-20 seconds per five or ten minutes, surprisingly little more can overheat them and cause internal damage to start.
And yes, the A4 like any old engine can benefit from a new ignition system. Given proper care and feeding an A4 can last an awfully long time. They have very low compression, which gives them relatively low power but very long life. You might check with some of the A4 gurus, I've been told the engine was shipped in two different styles, with one having a restrictor plate (under the carb) to lower the horsepower and lower the import tariff. Simply removing the plate and modifying the carb, boosts the power.