I think the basic test is to remove it from the carb, measure how far the needle is protruding, add 12V power to it and after a few minutes it warms up and the needle should extend a bit.
i'd do an internet search on 'how to test an auto bystarter' and you should get a ton of hits...OK, I just did and lots of results pop up.
Something else you could do is swap out the bystarter from your old carb to see if that one works with your new carb.