Quote Originally Posted by Throbert McGee
"Approximately how far is Cuba from the USA?"
Приблизетельно, как далеко находится Куба от США?
I would guess that "Приблизительно сколько километров от Кубы до США?" would be an equally common way of expressing the question. Just as "How long were you in Japan?" is often phrased in Russian as "Сколько дней (недель, месяцев, лет) вы были в Японии?", instead of saying "Как долго?"
Not quite. You're right about "Сколько" often used with time but it somehow doesn't sound as natural here. Or rather, it has different meaning.

"Приблизительно" in the question "Приблизительно, как далеко находится Куба от США?" is in fact somewhat redundant. What a form with "Сколько" isn't used already presupposes an approximate answer. (but you can add it to reassure the other speaker that he doesn`t have to look it up in an encyclopedia,)

If I ask "Сколько километров от Кубы до США?" or even "А сколько примерно километров от Кубы до США?" I expect somewhat more accurate answer.

Same with time:
"Как долго вы были в Японии?" - I have no clue how long have you been there. Even imprecise answer like "Несколько лет" would be OK.

"Сколько дней вы были в Японии?" - I know you were there for several days, but I need a clarification. You may use inversion to indicate you're not sure or use different units in your reply, but a number is expected to be given. "Пять дней." "Не помню, дней пять..." "Не знаю, пару недель..."