Alternator charges the battery.
Battery powers your electrical instruments.
CB or circuit breaker pops if the wires are getting too hot (means drawing too much current) for the CB amp limit on that CB, this is either due to wiring issue or the device it’s powering is acting up. The CB popping is to prevent a fire. You should let the wires have a few minutes (5+) to “cool” down before resetting and probably better to not reset it at all during flight as it means there is an issue (consider your situation, are you near an airport, good VFR weather, etc before your risk adding an emergency landing). It isn’t normal for CBs to pop in flight. I had this issue before and my mechanic put in a new alternator and I haven’t had the issue again until my fuel pump went out and it popped when I tried to run my broken fuel pump. So had to get a new one of those. Separate incident dates but now I have new ones instead of 40-60 year old ones and they have been working without issue.
Engine uses Fuel, Air and Spark to run
It doesn’t require electric for the engine to run on the Arrow, from what I believe at least, keep in mind your instruments may run on pitot static air, vacuum, or electric. Unsure if vacuum requires electric to run either if anyone knows?
Fuel is fuel flow, on low wing, the engine driven fuel pump is driven by the moving engine or the auxiliary backup electric fuel pump (you have a switch for that). On high wing, fuel flows by gravity but you still have the 2 fuel pumps.
Air is your induction air, if blocked you have alternate air. (Open / close). Alt air receives less air than induction air so your performance will drop a little bit (you can see this on the gauges during run up).
Spark is your magnetos, you have 2 mags (L and R), they are self contained units that gives spark to each spark plug in each cylinder. Each cylinder has 2 spark plugs, one controlled by the L mag and the other controlled by the R mag. That’s why if something is funky you can try to run on either the L or R mag, and also why during run up you do a mag check, the drop in RPMs show the loss of performance by using only one mag instead of 2 and that they both work independently of each other.
Keep in mind some aircraft do not have any electrical or battery in them, they start by hand propping. If you have engine out in flight, I’d say you might not have time to restart it. The wind will turn the dead engine prop automatically (unless you have feathering prop which is a multi engine thing), and that is acting as your starter, you just have to make sure you have fuel air and spark to start up. Given you have very limited time, and given my luck trying to start the engine after I fuel her up (aka a hot start), I’m not sure that I could do it successfully or not so I won’t try it. It would surely be a final test for your power off 180 if over an airport, and that’s easy to mess up depending upon the winds and luck.