I'm talking about topics such as model theory, proof theory, predicate calculus, Godel's incompleteness, ordinal numbers, large cardinals, etc. How much of these do I need to know if I want to specialize in some other areas of mathematics such as algebra and topology?
None or virtually none, as far as I know, just like you don't need to know any complex analysis to be a good algebraist. You'll need to know the basics of cardinality (uncountable versus countable, finite versus infinite) and ordinals (there are some situations in algebra and topology where it's useful to have sequences of length $\alpha$ for an arbitrary countable ordinal $\alpha$). Other than that, none of it is critical knowledge for other fields.
However, speaking as someone who specialized in logic, I'd say that the more logic you know the better a mathematician you'll be - a firm understanding of foundations will help with whatever you're doing. Model theory and algebra have enough commonalities that, depending on your sub-specialization, it would probably be useful for an algebraist to have some model theory. Similarly, topology and set theory have some connections that would probably make it worthwhile to study set theory if you want to go into topology.