Agüito comes from the verb agüitarse, which is a colloquial Mexican Spanish term meaning 'to get sad', 'to get down', or 'to feel discouraged'. It's a more informal and expressive way to say me pongo triste.
In the song, Christian Nodal sings, "Y yo no soy nadie y pos' me agüito" (And I'm nobody and well, I get sad). This word perfectly captures the feeling of dejection and self-pity the singer experiences when he feels he's not good enough for the person he loves, making it a relatable and emotionally resonant choice.
Everyone around him has already delivered the verdict. "Dicen que ya ni me clave" (They say I shouldn't even bother), and the reasons stack up behind it: "Que pa' mi eres inalcanzable" (That for me you're unreachable), "Y yo no soy nadie y pos' me aguito" (And I'm nobody and so I get bummed out). Nodal does not argue with any of it. He concedes the whole case out loud -- "Puede que sea cierto" (It might be true) -- and then declines to act on it.
What he actually asks for is smaller than love. He wants to be noticed at all, "Esperando a que me peles" (Waiting for you to notice me), and not to be introduced to her friends as "tu amiguito" (your little friend). The dare in the chorus, "Anda calame tantito" (Come on, try me a little), is as close as he comes to pressing her. Everything else is just refusing to move: "Aunque me trates como un perro / Aqui sigo tras de ti" (Even if you treat me like a dog / I'm still here after you). The song never resolves that standoff, which is the point -- he is still standing there when it ends.