Am I eligible to go to Australia given my age, or is New Zealand a better option for a Master's?

Both Australia and New Zealand have no upper age limit for student visas, so you are eligible for both.

However, if you are over 35, both countries may look more closely at how your chosen course fits your career goals and post-study plans. New Zealand is generally seen as more flexible for mature students, and your budget and timeline align well with typical New Zealand Master's programs.