Specifically methylates the cytosine at position 1962 (m5C1962) of 23S rRNA. cytidine(1962) in 23S rRNA + S-adenosyl-L-methionine = 5-methylcytidine(1962) in 23S rRNA + H(+) + S-adenosyl-L-homocysteine Belongs to the methyltransferase superfamily. RlmI family.