The expressions "al día" and "por día" are almost equivalent.
We use "a + (period of time)" or "por + (period of time)" to distribuite a quantity in a period of time.
Usually both are equivalent, but is preferrable using "a" for long time periods (day, week, month, year) and "por" for short time periods (week, day, hour, minute, second).
Voy al cine dos veces al mes.
Me gusta ir a la playa una vez al año.
Este coche alcanza una velocidad de 100 km por hora.
El motor de mi coche funciona a 3000 revoluciones por minuto.
With "día" and "week" you can use "a" or "por" as you prefer.
Estudio inglés dos veces por semana.
Bebo agua cinco veces por día.
> Rach_US 10 minutes ago
> So, if I´m at a hotel and want to ask how much it costs per day, can I use
> both "Cuánto cuesta por día?" and "Cuánto cuesta al día?"
You can use the expression you want, but remember that in Spain we dont ask the price by day, but by night ;) "Cuánto cuesta por noche?"