You got me curious enough to consult the OED. It's an American colloquialism:
B. noun (elliptical uses of the adj.) ... 5. Temper; passion. orig. U.S. and dial. 1834 D. CROCKETT Narr. Life iv. 30 Her Irish was up too high to do any thing with her. 1860 BARTLETT Dict. Amer. (ed. 3) 217 My friends say that my Irish is getting up, meaning, I am getting angry. 1877 F. ROSS et al. Gloss. Words Holderness 80/1 Iry; Irish, E. and N., passion; anger; rage; fury. 1933 PARTRIDGE Words, Words, Words! I. 9 Both Irish and the colloquial Paddy are used for anger. 1949 R. HARVEY Curtain Time vii. 73 But George's Irish was up. 1972 Evening Telegram (St. John's, Newfoundland) 23 June 1/4 ‘I got my Irish up,’ he said, ‘and here's a man that's going to fight back.’
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OED
Date: 2009-02-07 07:15 pm (UTC)B. noun (elliptical uses of the adj.)
...
5. Temper; passion. orig. U.S. and dial.
1834 D. CROCKETT Narr. Life iv. 30 Her Irish was up too high to do any thing with her. 1860 BARTLETT Dict. Amer. (ed. 3) 217 My friends say that my Irish is getting up, meaning, I am getting angry. 1877 F. ROSS et al. Gloss. Words Holderness 80/1 Iry; Irish, E. and N., passion; anger; rage; fury. 1933 PARTRIDGE Words, Words, Words! I. 9 Both Irish and the colloquial Paddy are used for anger. 1949 R. HARVEY Curtain Time vii. 73 But George's Irish was up. 1972 Evening Telegram (St. John's, Newfoundland) 23 June 1/4 ‘I got my Irish up,’ he said, ‘and here's a man that's going to fight back.’