The person is declaring that he/she will
もう's literal meaning is "has gotten to the state" (e.g. もう歩けない、もう食べれる), and here it indicates that the speaker has gotten to the state that he/she will never board a train or car (he/she "had enough").
は specifies that he/she will specifically not board a train or car (while he/she might board other means of transport, like a ship or plane). It has the effect of singling out trains and cars.
乗らん is short for
乗らない and shows some aggressiveness. Here it shows that he or she had enough of trains and cars and is angry enough to vow never to board them again. (e.g. saying いらん、やらん sounds much more rude/strong than いらない、やらない）