First off, don't be discouraged, its not as difficult as you think.
However, you want to make sure the rifle is a post 285,507 serial number. The earlier, low numbers can be dangerous to shoot.
Without seeing the rifle now, I'm going to assume you need a stock and barrel.
The CMP sells both. If your rifle has the rear sight (1903) then you are a head of the game.
The stock wont take much fitting, it will be easy. It should come with the upper wood.
PM me with your E-Mail address and I'll send you the TM for the M1903 (in PDF format) that will show you what parts you need and how they go together.
You can get any missing parts from Sarco or Gunpartscorp.
The only difficult, (not really difficult but you do need a Chamber Reamer and headspace gages) is installing the barrel.
Its not hard. You'll find a mark on the receiver, and one on the barrel. Screw the barrel on until the marks line up. It should be "short-head spaced" meaning you'll have to use a reamer to take a few thousand's off so the Go Gage Works. This should be done by hand, not a lathe. You wont be removing much metal so be careful. If you haven't done any chambering before you might want to take it to a gun smith. It wont cost much, it just takes a couple minutes.
If you are going to shoot it I would highly recommend you true up the face of the receiver with the internal receiver threads, you'd be surprised how this will improve accuracy.
Don't let the above comments discourage you. I've done several, its not that hard. Except for the barrel chambering, its real easy, just a matter of putting it together.
I would recommend against the Lyman sights if you want it back too the M1903.
It the holes drilled for the scope mount bother you use plugs to cover them up.
Anyway if you need help with this project pm me, I can get you going. This will make an outstanding project and rifle to pass down to your kids.
However, Friday morning I'm going on a spring bear hunt and probably wont be able to get back to you for a week to ten days.
Just about all M1903s are rebuilds. I don't get concerned with "all original' They are as rare as hens teeth.
This is my personal M1903A4, all made with parts, it wasn't original. But it meets all the rules to the CMP Vintage Sniper Matches.