Hooray Henry

A loud-mouthed but ineffectual upper class fool/a hunting-shooting-fishing type. This sounds very much like a phrase of British origin - possibly from the pen of P. G. Wodehouse. In fact it's an American phrase, coined by Damon Runyon. In a piece for a 1936 edition of Collier's Weekly, Runyon wrote:
"He is without doubt strictly a Hoorah Henry, and he is generally figured as nothing but a lob [slang word for useless idiot ]as far as ever doing anything useful in this world is concerned."

The female counterpart is a Sloane Ranger, like Lady Diana Spencer. See http://www.phrases.org.uk/meanings/sloane-ranger.html. Sloanes are widely derided and, regardless of whatever educational standing they may have achieved, are generally regarded as dim-witted socialites.:twit:

Off the top:
For Sloanes (female in particular), the first thing that sprang to my mind while reading the phrases.org webpage - because the term was derived from Sloane Square - was Κολωνακιώτισσα, the one who lives in the vicinity of Kolonaki Square, a traditionally upper class and expensive district near the centre of Athens.

Some of those types have moved to the northern suburbs - along with quite a few of the rich and nouveau riche - and another term was subsequently coined, βορείων προαστίων, which can be used with κυρία or τεκνό (along with its acronym: ΤΒΠ).

Another term - broader, Ι believe - would be γόνος καλής οικογενείας (used both for male and female, often mockingly).

And I'm certain all that will be duly screened, analysed and categorised by our fellow Lexilogians.

I'd stick to the first part of your definition — you won't find many hunting-shooting-fishing types here, more the clubbing-boating sort.
I think πλουσιόπαιδο would be a rather innocuous translation, with none of the silliness involved.
Κολεγιόπαιδο can be used as a term of derision.
Λελές might be a closer equivalent.

And now that I've thought of λελές, you can continue here:
http://lexilogia.gr/forum/showthread.php?393-fratty-κολεγιόπαιδο-λελές

PS: Talking of ΒΠ: παιδί των βορείων προαστίων.

...
Monty Python's The 127th Upper Class Twit of the Year Show (Hooray for Henry Smith-Smythe-Smith, Vivian's brother)

Sloane Rangers in Wikipedia and some of them in Lexilogia.

Από τζάκι.