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.
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.