Do I need a new alternator or just a jump start?

Jump start it first. Turn the lights, wipers and fan on. If it doesn't die when the jumper cables are removed then the alternator is working. Use a multimeter across the battery terminals (if it doesn't die.) If you get 14v or better the alternator works and either the battery or the connections (including the engine ground) are bad.

Try jump starting before you spend money on a new alternator

Blame battery first. Batteries will last 5 to 7 years, most going the full 7, but lets say in your case they one did 6 and 2 did 7. That adds up to 20 years.
Car is a 1999 +20=2019 Replace the battery. A new 72month battery is about $100.00 as they also want the old one(the core) and they give you a few bucks for that so it comes off the purchase price for a new battery.
. My battery was a $200 battery(which had a service life of 10 years). As soon as there was indications that the car would not start....the machine gun fire from the starter.....replaced the battery for a cheaper $100 battery.
Hardly batted an eye. Sure I could have dragged another year out of the old battery, but EVERYTIME I WOULD HAVE TO BE JUMPSTARTED...that is so inconvenient. You also stand a chance of burning out the alternator and the starter motor WHEN YOU HAVE A WEAK BATTERY. A starter and an alternator is about $400.00 EACH. so if a mechanic was doing the work you are looking at a cool thousand.(or 10x's the cost of a battery)Alternator $400 + starter$400 + battery$100 +labor=$1000+

Your choice. You have been informed.
Personally, I would park the car until I can afford the battery and swap out the battery in my yard...owning a single 1/2" open end wrench. Done.
(Had buddy drive me to store to buy another battery. Later drove my car back and gave them the battery core for the discount.
Done.
The old battery is recycled and made to perform like new.

The warning light for the charging system is separate from the light set by most trouble codes.
In your case, start by attempting a jump. If the car starts, and stays on after the cables are removed, I recommend driving directly to somewhere that can read the codes AND load test the charging system. Most parts shops can do that.

How about that bright red battery charge warning light?
Charge the battery with a portable battery charger set to 12 volts, 10 amps, for 6 hours.
Then start the car and drive directly to an auto parts store and they will check the battery and charging system for free.
If the alternator is bad you'll have around 20 minutes of drive time so keep all you electrical equipment off.
If the alternator is bad you won't get more then a miles before you get stranded if you get the motor started so I would't drive it.

If the battery is 4 years old get a new one regardless. Then check the alternator with a voltmeter. If not old then charge it with a battery charger. Jumping and charging old dead batteries is hard on pos car batteries like yours. I would not advise that. If the alternator checks bad then change it. No reason to guess here or ever. Excuse my bluntness.

Try jump starting it.
Let it run for awhile (30 minutes to an hour)
If you shut it off and it goes back to the ticking mess, then either the battery isn't holding a charge or the alternator isn't charging the battery.
Also, make sure that nothing is drawing power while the vehicle is off. You'd be surprised at how many lights there are in a vehicle and some of them can inexplicably be on when the vehicle is shut down.

To begin with, you need a charge. Then have the charging voltage measured. This is the sole test. If it is not there and the belt is on the alternator then it could be the connections or the alternator itself. If you ever take the alternator out you could have it tested free at Autozone. For that matter, you could have it tested in the car for free there too.

The first thing that I would do is get it started and measure the battery voltage at idle with the headlights on and no battery charger or jumper cables connected. Then I would know something. So if you do it, then YOU will know something.

-General automotive mechanic since 1972

Stop trying to think, Skippy. It's obvious that you're not properly equipped for the task. If your car refuses to start, you have a DEAD BATTERY. It's not always caused by a bad alternator.