Learn English – Is “anytimes” actually a word

Using "curious" in the sense of "odd" or "strange" is more British than American. It does still crop up here and there, but is somewhat archaic, formal -- or, in fiction, intended to convey a British and/or old-fashioned kind of person. E.g., 'How curious!' Dorothy said, peering at the bright purple tree.

So absent context, "The curious boy" will be ambiguous. No getting around it. You have to tell whether it's about the boy investigating something ("The curious boy pressed deeper into the forest") or about a boy being odd in some way ("The curious boy walked down the sidewalk, and we all stared at him"). Or if it means both!

This ambiguity will also be used when someone wants to make puns. "What a curious creature you are!" may mean "you are always wanting to investigate" or it may mean "odd." I recall there being some of this double-meaning in Alice in Wonderland.

"He's very curious" will generally be understood to mean someone has the quality of curiosity, not oddness, in modern American English, but if you want to reduce ambiguity, you use "about" to clarify. "He's very curious about everything!" "She's very curious about ancient Egypt." "The neighbors very curious about our business, and I want better curtains in our house!"

In modern American, "curious" people or animals (the curious cat, the curious child) will be "inquisitive" with an ambiguous side-order of "odd"; the more stereotypically inquisitive the animal (or person: the curious neighbor), the more likely the assumption will swing to "inquisitive." But "curious" inanimate objects will always be "odd." And "a curiosity" is an odd thing.

(You wouldn't use "Curious you!" because either way, applying adjectives to pronouns rarely works well. "Red you!" "Angry him!" "Stinky her!" "Inquisitive me!" -- you can get away with it sometimes, in a slangy fashion, but you tend to need a noun, like "Curious George," the monkey who is curious about everything! (The Curious George books are a series of children's books which strongly normalizes "curious" as an adjective meaning "inquisitive.")

So if you want to avoid ambiguity, use "odd" or "peculiar" or "weird" or "fascinating" in American English, to mean those things. And only use "curious" for things like "He's very curious" (inquisitive) or "I'm curious about this sentence here" (I want to understand this sentence better, I am interested in this sentence).

For the first part, I would be inclined to use "I am free" unless there's a specified point at which you will become free, e.g. "After lunch, I will be free for the rest of the day".

For the second, you certainly can use "today" instead of "the day". This is just preference, but "the day" sounds more natural unless you want to specify another day e.g. "I'm free the rest of today, or tomorrow morning".

"Yesterday" also works too, but sounds a bit clunky in your example. I think it sounds more natural without the "rest of" - so simply "After that, I was free yesterday".