As a rough rule of thumb good alternators can usually keep up with a duff battery when the car is running, duff batteries usually cause a problem running down on charge over night (wont crank over after being stood).


as your car starts fine from being stood then packs up shortly after I would guess the car is using everything the battery has to offer and the alternator aint putting it back in (not charging)


alternator warning lights don't always come on, if you don't have access to a battery drop tester then im sure your local garage would be happy to check your battery for free as they could potentially sell you a new one, just keep an eye on there results and ask that they show whats going on.