Hello!
Yes, you do damage when the spike-ball hits the yellow shield. The number you're looking at is the amount of times you have to break her shield to beat her. It's the number after the 2, that is the current health of her shield. (You can change spike-ball with right mouse click or number key 1 to 5)
As for the upgrade, it's not the gold is used to upgrade. It's the mana itself. The gold is used to buy stuff from the shop in the main menu. So from start of the game, you have 20 mana to upgrade something equal or fewer mana.
As to defeat the first enemy, one way (Gaining gold ASAP) is to upgrade the 'mana regen+' for 20 mana. This will gives you 2 mana every cooldown. Once you get 5 mana, upgrade 'Damage+' once. (For every damage, you get one gold). After that spam spike-ball before she throw a dart (You need to time it as best you can). Now, every time the ball hits the yellow shield, you get 2 coins. If by chance the dart misses the dart-board. There will be a green symbol that appear in the middle, click this as fast as possible (You only have a few seconds) . This will spam random free balls.
Once you lose, you should have some gold, go to the shop and buy 'starter mana' (This will give you more mana to play with for every game). Then repeat. The idea is to slowly upgrade yourself until you can beat the enemy.
You can beat her without gong to the shop, but that takes a while of grinding.
Thanks for the reply, I'll fix the issues you mentioned and update it. Will also make a more expanded game mechanics. Thanks for playing!