Well, I gave it a shot, and the translation came out roughly the same as the one already posted on Youtube, just as I expected. Here it is, along with the transcription from the video, and the other translation.
ITALIAN VOICE OVER
Devo fuggire. Devo fuggire!
Non so cosa troverò fuori, ma devo fuggire.
Forza, forza! Più forte!
Se non regge sono finito...
Il motore della macchina che doveva portarmi verso la libertà sta per lasciarmi. Prima che il mio corpo sia disintegrato dai miei inseguitori, voglio lasciare un messaggio telepatico, perché i miei posteri possano conoscere la mia vera storia.
La storia iniziò nel lontano luglio del 2002, duecento settanta secoli fa. Alcune radiazioni solari determinarono la deflagrazione contemporanea di tutti gli ordigni nucleari costruiti dall'uomo.
I pochi superstiti si rifugiarono in profonde caverne, unici luoghi non contaminati dalle radiazioni, e là iniziarono la ricostruzione del loro mondo. Uomini di tutte le razze giurarono solennemente che mai più ci sarebbe stata guerra, e si dedicarono alla costruzione di un mondo perfetto.
Ma la perfezione non è dell'Uomo. La perfezione è solo nella Macchina costruita dall'Uomo, perciò era la Macchina che doveva regolare la vita dell'Uomo, e l'Uomo costruì il Robot. L'Uomo costruì Robot sempre più perfetti, e il Robot fu tanto perfetto, che cominciò a costruire l'Uomo, e a controllarne ogni suo atto.
L'Uomo voleva un mondo dove fosse bandito il Male e l'Infelicità, e il Robot costruì un Uomo asettico, scevro dal Male e dall'Infelicità.
Ma l'eliminazione del Male è l'eliminazione del Bene, in quanto l'uno è la contrapposizione dell'altro. Già l'Essere Supremo aveva costruito un Paradiso Terrestre, nel quale l'Uomo visse, finché non sentì che solo avvicinandosi all'Albero del Bene e del Male, avrebbe trovato il Male, ma anche il Bene e la Felicità.
Non so perché questo sentire, in questo mondo sia toccato a me, perché ho dovuto cercare questa Felicità che sta portando il mio corpo alla disintegrazione. Forse il mio nome non è quello posto sulla provetta nella quale fui costruito, THX 1138. Forse il mio vero nome è Adamo.
ERRIWAN'S TRANSLATION
0.31 I have to make it, I have to escape!
0.33 I don't know what I will find out there but I must escape
0.39 Come on, come on! Faster!
If it (the car) doesn't hold up I'm finished
START OF THE OFF-SCREEN NARRATION
2.00 The engine of the car that was leading me to freedom is going to die soon. Before my body is disintegrated by my stalkers I want to leave a telepatic message so that posterity will know my real story.
2.19 - The story begun in the distant July of 2002, 270 centuries ago. Solar radiation determined the explosion of all nuclear heads all over the World. The few survivors took refuge in deep caves, the only uncontamined places... and there they started to rebuild their World. Men of all races sworn that never again was to be War and they dedicated their efforts to rebuild a perfect World. But Perfection doesn't belong to the Man. Perfection is only in the Machine created by the Man, therefore it was the Machine which was to regulate the life of the Man... so the Man created the Robot. He created it more and more perfect and the Robot was so perfect that started to create the Man and to control him in all his acts. The Man wanted a World were Evil and Unhappiness were banned and The Robot created an aseptic man, free from Evil and Unhappiness. But the elimination of Evil also means the elimination of the good, since one is the opposite of the other. Already once the Lord created an Eden where the Man lived until he felt that only approaching the "Tree of Knowledge of Good and Evil" he would have found Evil but also Goodness and Happiness. I don't know why in this World it happened to be me and why it was me who had to look for the Happiness which will lead to my disintegration. Perhaps my name is not the one placed on my test-tube in which I was created, "THX-1138", perhaps my true name is Adam.
MY TRANSLATION
I have to escape. I have to escape. I don't know what I'll find out there, but I have to escape.
C'mon, c'mon, faster....
If this thing doesn't hold up I'm done for.
The engine of the car that was going to carry me towards freedom is going to fail soon. Before my body is disintegrated by my pursuers, I want to leave a telepathic message so that my posterity will know my real story.
The story begun in the distant July of 2002, two hundred and seventy centuries ago. Some solar radiations caused the simultaneous explosion of all nuclear warheads built by Man. The few survivors took refuge in deep caves, the only places uncontaminated by radiation, and there they began to rebuild their world. Men of all races solemnly swore that there would be no more wars, and dedicated themselves to building a perfect world. But perfection does not belong to Man. Perfection only belongs to the Machine built by Man. So the Machine had to regulate the life of Man, and Man built the Robot.
Man built more and more perfect Robots, and the Robot was so perfect, that it began to build Man and to control every action of his.
Man wanted a world were Evil and Unhappiness were to be banned, and the Robot built an aseptic Man, free from Evil and Unhappiness.
But the elimination of Evil is the elimination of Good, because one is the opposition of the other. Already the Supreme Being had built an Earthly Paradise, in which Man had lived, until he felt that only approaching the Tree of Good and Evil, he would find Evil, but also Good and Happiness.
I don't know why this feeling in this world had to get to me, and why I had to search for this Happiness that is leading my body to disintegration.
Maybe my name isn't the name that was on the test tube in which I was built, THX 1138. Maybe my real name is Adam.