I know it is better to just say before noon, but if you wanted to say "before the afternoon arrives" or "before the afternoon ends" would it be correct?
What the fragments "before the afternoon arrives" and "before the afternoon ends" do is make afternoon into something which can act, which can do something.
While day can break and night can fall, morning and afternoon don't generally do anything; they just are. Consequently they don't end or arrive, and one says "before the end of the afternoon", or "before the end of the morning".
Because the use of before sets a deadline, it's usual to align it with the end of a period, not its beginning, so one wouldn't generally say "before the start of the afternoon", but "before the end of lunch".
Following Kris's comment, perhaps the question needs to be explicitly answered: Before the afternoon arrives or ends is grammatical but is certainly not idiomatic. Using those expressions is not generally to be recommended.
"before the afternoon arrives" or "before the afternoon ends"
There's nothing ungrammatical about them, but these expressions sound rather unnatural to my native ear.
As to a better way to word it, that depends on the context. Am I talking to a coworker, about when we should meet with a client? To the appliance store, about when they are going to deliver my new washing machine? To some friends, about when we are going to meet for drinks?
In the workplace, I think "by the end of the afternoon" might sound more natural than "before the afternoon ends." For example:
"Hey, Boss, when do you need that report?"
"If I could get that by the end of the afternoon, that'd be great."
(In this case, by the end of the afternoon essentially means the same as by the end of the workday, or before you go home today.)
If the appliance store was going to make a home delivery, I'd expect them to say something more like:
We should be there before noon.
We should be there before the afternoon arrives.
As for drinks, I really don't expect I'd ever suggest to my friends:
Let's meet before the afternoon ends.
Instead, I might say one of these:
Let's meet around four (o'clock).
Let's meet right after work.
Let's meet late afternoon/early evening.
Let's meet before suppertime.1
I'll offer one more example. Say we're planning a picnic; I might say:
Let's start around two. We'll want to wrap things up before it gets dark.
seems more natural than:
Let's start around two. We'll want to wrap things up before the afternoon ends.
1 (or dinnertime, but that's another question...)
"Before noon" is non-specific and therefore often meaning any afternoon, but not always. It's also less specific in terms of the timeframe, not implying any hurry. The phrase "before the afternoon arrives" carries some urgency. Even though it doesn't specify exactly which day; could refer to any day; it specifies it's the day, adding more impact or urgency.
This is from my understanding of the phrase in common usage, not a scholarly appraisal of these related phrases.