In the example in the headline to this question, the statement qualifying the question makes the punctuation seem ill-fitting...
Here's another example:
Will you be able to make a decision on this, as it’s getting on to a year old?
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
This is a punctuation/style question, so the relevant guidance amounts to recommendations that may or may not be consistent from one style manual to the next. If you like (or have to use whether you like it or not) the Chicago Manual of Style, you'll find decisive guidance in section 6.71 of the fifteenth edition (2003):
6.71 Within a sentence. A question mark is used within a sentence at the end of a direct question [cross reference omitted]. If the question does not begin the sentence, it need not start with a capital letter [cross reference omitted].
Is it worth the risk? he wondered.
The question, how can the two be reconciled? was on everyone's mind.
The Oxford Guide to Style (2002) reaches a similar conclusion on this point:
5.8.1 Typical uses [of the question mark]
Matter following a question mark begins with a capital letter
Do you want more ladycake? Buns? Muffins?
You will be back before lunch, right? About noon? Good.
but questions embedded in another sentence are not followed by a capital:
Where now? they wondered.
He pondered why me till his head hurt.
And Words Into Type, third edition (1974) makes three:
Place the question mark at the end of the question, not elsewhere.
Poor: Will she fight Victor's case, I wondered?
Better: Will she fight Victor's case? I wondered.
Wrong: "It's a might hard pull, though, isn't it? when people don't have confidence in you."
Right: "It's a might hard pull, though, isn't it, when people don't have confidence in you?"
In this last instance, I gather that WIT considers placing the question mark immediately after "isn't it" to be wrong because the question isn't fully asked until the "when" clause has run its course.
Applying the recommendations to the OP's examples
In both of the OP's examples, though, the question seems to be complete in the middle of the sentence, and I believe that Chicago, Oxford, and WIT would all endorse the following punctuation:
How should a question ending in a statement be punctuated? as ending it in a question mark seems a little off.
Will you be able to make a decision on this? as it’s getting on to a year old.
Of course, you usually have the further option of breaking such constructions into separate sentences, unless you're working with transcriptions of speech. For example:
How should a question ending in a statement be punctuated? Concluding with a question mark eleven words after the direct question is finished seems a little off.
Will you be able to make a decision on this? It’s getting on to a year old.