"Solid Jacket" is another term for "Full Metal Jacket." Remington uses the term "metal case" to mean FMJ, which makes even less sense. Whichever term is used, this means that there is no exposed lead at the nose of the bullet.
Bullet jackets start out as a cup, with the lead core dropped in and the jacket formed around the core in a die. Either the nose or the base will have exposed lead. Obviously, soft points and hollow points have the jacket opening at the top (so do "ballistic tips", which are just HP bullets with a plastic plug in the tip). FMJ bullets have the opening at the bullet base.
An exception to this is plated bullets, which will fully coat the entire core.
None of this has anything to do with AP ammo. AP ammo uses some kind of hardened core to enhance penetration of metal. Most bullets use a hardened steel core, but recently, there have been specialized rounds that use tungsten as a core. Tungsten is very expensive, but not only is it hard, it is also denser than lead, so a given bullet weight can be made shorter by substituting tungsten for lead. The only 5.56 AP is the M995 round, which uses tungsten for the core. This ammo costs the military over $1 per round, so it is rarely used, and is only available in SAW belts as a special-order item.
And, no, none of it is available in any form on the civilian market, nor is it likely to be in the forseeable future.
-Troy