RB, welcome. I have a '77 C-30 also. I knew jack-$#!t about an A-4 when I got the boat 10 years ago.
We can help you through it. We will get into some deep details, but they are all important. As Al noted, fuel lines, filter(s), what the PO did or did not do, elec/mech pump, etc.etc.etc. Be patient with the questions, we will help find the answers!
One comment on the video - There is no real neutral..the whole assembly is a friction plate/clutch pack thing, so sometimes "neutral" will drag the shaft..next time try the shift lever a little more aft...I have found on mine, "neutral" has a little whine in the gear assembly..you just need to find your sweet spot..do not be concerned with this for now.
Why it won't keep running is the next step!
- Some more videos and pics of the fuel system will help. Most of the fuel system is on the port side of the engine, so let's have a look! What we are looking for is whether you have an electric or manual fuel pump, the condition/age of the fuel lines, are there any additional filters (the 77 C-30's only came with a pitiful screen in the mech fuel pump, a lot of us have added additional filters) does the boat you have a fuel pressure gauge (added by the PO, not stock) and other things. Each of these write a page in the story of the fuel system, and we will work thru it with you!