Apuro translates to 'trouble' or 'predicament'. It's an interesting word because it describes a difficult situation, but often one that can be resolved or escaped from.
In the song, Rusherking sings, "Yo soy de los formales que te sacan del apuro" (I'm one of the formal ones who gets you out of trouble). He uses this to contrast himself with other men, portraying himself as reliable and someone who will help his love interest when she's in a bind, rather than cause her problems.
Rusherking's "Cuéntame" is a sweet and direct song where he's basically saying, "Tell me everything, even though I feel like I already know you!" He's making it clear he's not like past partners who left too soon. He promises to love his person forever, saying, "Te amaré y estaremos los dos / Por siempre miraré tu rostro" [I will love you and we will be the two of us / Forever I will look at your face]. He even dreams of taking them on adventures, from "Argentina a Boston" [Argentina to Boston]!
He wants to make his intentions super clear, telling his love, "Yo soy de los formales que te sacan del apuro" [I am one of the formal ones who will get you out of trouble]. He's a bit blunt, but honest, about other guys, saying, "hay hombres que sólo quieren tu culo" [there are men who only want your ass], but he wants to make her happy. He's not promising fancy things like "cartera y zapato" [purse and shoes], but instead, he's offering experiences: "Yo te prometo ir al mar y pasar un buen rato / Bailando contigo, volar muy alto / gastar en viajes lo que gano en contratos" [I promise you to go to the sea and have a good time / Dancing with you, flying very high / spending on trips what I earn in contracts]. It's a charming promise of love and adventure over material possessions.