"traditional" approach (rather than using romaji for everything)
Would certainly recommend this - romaji may seem easy and convenient but it's really quite hard to wean yourself away from; it's next thing to useless in both cultural and communicative terms (Japanese don't dislike romaji; they just don't use them for anything more than proper names. A bit like writing English in IPA 
)
IIRC to be able to read a newspaper one needs around 1,900 kanji
But you can go a long way towards making sense of one with fewer than half this number 
As said, romaji is only really used for proper [western] names where a katakana hasn't been adopted. You wouldn't normally use romaji beyond that - other than for computer data entry where other options don't exist.
One of the Japanese printers I've worked with once claimed there were potentially 50,000 kanji characters, but Japanese typesetting systems cover 10,000. The official list (as taught to Japanese schoolchildren) is about 2000. You will indeed manage fine with knowledge of a 1000 - probably a lot less if you don't mind reaching for a reference book periodically.
Of course, this assumes you know enough hiragana to make sense of the kanji. Otherwise it's a bit like having random names and phrases shouted at you.
