E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
Monkey = the Devil;
an imp of mischief. Hence, a meddlesome child is spoken to as you little monkey; and is called a regular imp, or imp of mischief. The allusion is to the old drawings of devils, with long tails and monkey ugliness.
1
To get (or have) ones monkey up. To be riled. Here the allusion is also to the devil or evil spirit in man; he will be in a devil of a temper. Even taken literally, monkeys are extremely irritable and easily provoked.