I think there two fundamental points that you should amend in your plans
1) consider the MTR. It's clean, efficient, fast and opens your choices tremendously
2) give yourself more than a few days before you lock in an apartment. That way you will very quickly experience all the points mentioned here about travel distances, type of apartments, what they do and don't have, etc
So I would rather arrive 2 weeks or 1 month early, live in a crammed but affordable place until you have locked something in and move properly. The long term gains on this approach are substantial so it's worth the initial discomfort.
For affordable places that suit you, don't go to East hotel or so as suggested (expensive for a whole month) but consider guesthouses listed on Hadla.gov.hk for affordable hostels or something like the ibis hotel in North point. I remember they used to have a 10k/month deal going some time ago