Adding a cap does not save the alternator or the battery, it stores power for instantaneous power on bass heavy passages. Gives the power supply more even headroom.
The alternator will fail if it is undersized for the load. If the car has power accessories then it will most likely have an 80 or 100 amp alternator which is sufficient. Music does not produce a constant current draw, it is in short bursts. The alternator is what powers everything when the engine is running. Adding extra batteries is just for extended listening with the engine off before the batteries die. Adding extra batteries also adds extra load to the alternator so it has more charging to do.
The amp is probably only 200-250 watts rms clean. The 1000 watt rating is the ISL rating (if struck by lightning). The rule of thumb is about 100,000µf of capacitance added to the power supply for every 100 watts of amp output power. So that amp would be fine on a .25f or 250,000µf capacitor. Mount it as close to the amp as possible for best results as it will have the lowest voltage drop to the amp when needed.
It does not need to be fused but make sure that you install a fuse no further than 18" from the battery for the amp power wire so if there is an accident or short circuit it will not burn up the carpet like a dynamite fuse or set fire altogether. a # 4 wire will be plenty. Don't forget to upgrade the ground from the battery to the car body with the same size wire. This is just as important as the amp will be grounded to that car body and used as the ground so it needs to be the same size as the hot wire for best performance.
Run the power wire and turn on wire down one side of the car and the signal wire down the other to eliminate noise induced into the system. Mount the amp on a non conductive surface, don't mount directly to the car body.
As said before, use sound deadening materials. A good alternative is electrical duct seal. Buy a bunch of clay bricks of it and add it to the backside of all of the door panels and any panels that rattle. This will stiffen up the car and make the panels that the door speakers are mounted in more like a speaker cabinet.
Set crossovers and gains accordingly. Also, If the box has 4 ohm speakers then just run it in stereo mode. If you bridge it and parallel the speakers then each side will see half the impedance and it will be a 1 ohm load which is too low. If you bridge it you would need two 8 ohm speakers in parallel to achieve a 2 ohm load bridged. For a safe bridged load with the 4 ohm speakers you have now you would have to run them in series and that would actually decrease the output from the stereo mode alternative with what you have. The amp will play cleanest in 4 ohm stereo. You can sum the inputs for mono bass since it is non directional.
Avoid going above 100-120Hz on the crossover frequency to avoid skewing your front stage. You want to create the illusion that all of the sound is coming from the front of the car so you install tweeters only in the front with midrange and then midrange only in the rear for rear fill. It will take some time to get all of the gains and crossover frequencies set to where they sound great so go easy a little at a time and audition music that you are familiar with to help you set things right.
