Do I have to discard Buffer RLT with beta-Mercaptoethanol (ß-ME) added to it after 1 month of storage?

No. Beta-Mercaptoethanol (ß-ME) is stable for 1 month, but Buffer RLT itself is stable for at least 9 months at room temperature (15 to 25°C). Simply add fresh ß-ME to the Buffer RLT supplied in RNeasy Kits to ensure complete inactivation of RNases while isolating RNA.