From here on, I will call the two things 'rock' and 'sun'.
We still call things going very fast as being in an orbit. We call these hyperbolic orbits, and this orbit carries the rock away from the sun after a single pass. These items have 'escape velocity'
Anything at less than escape velocity will be in a normal orbit. A normal orbit is not circular, it is an even, oval shape called an ellipse. The earth's orbit is nearly circular. We say that the earth's orbit has 'low eccentricity'. Its speed doesn't slow down or speed up much over its orbit. It stays at around 30 km/sec all year. A comet's orbit is highly eccentric. When it is far away from the sun, it moves very slowly. Then it falls toward the sun, gaining speed. But that little bit of movement it had caries it away from the sun, so it misses the sun. The sun's gravity pulls the comet around, flinging it back out where it came from. It slows down again as the sun's gravity pulls it back, until it slows down, turning, and falls back again.
So, let's examine your statement that "anything that approaches the orbit of any body in space , will be drawn in to that body by virtue of gravity, unless it has sufficient mass and momentum to maintain an orbit." There is something here that has created this understanding, the understanding that anything that is moving will be stopped by friction. Any movement soon stops. But what is there is space to make a rock stop moving? There is no air to slow it down, no carpet to rub against. Only if it happens to run into something else, and there is not much in space to run into.
Anything that approaches a body in space will be in an orbit.
There is also the understanding that small things stop easier than big things. But this is again tainted by the fact that your life has been lived on a planet, with air things have to push through and surfaces things have to rub against. Forget those things - there is neither in space - and small things keep going just as well as big things. The little thing has less momentum, but it also has less weight - the force of gravity on it - and the two things cancel out. In all orbital equations - as long as the 'rock' is much smaller than the 'star' - the mass of the rock cancels out, and is irrelevant. For our original premise - which was the idea of launching a payload of nuclear waste into the sun, by the way - the payload of waste and the earth follow the same equations - their wildly different masses are irrelevant.
What about when you throw a stone at your brother? That stone is an object near the earth. Surely it isn't in an orbit? Well - it is. It follows an elliptical path like anything else. It is like that comet, moving fairly slowly far away from (the center of) the earth, and it would continue in an orbit unless something - your brother, a window, the earth itself - gets in the way. If you could throw it at 8 or 9 kilometers per second (and if there wasn't any pesky air) it would remain in orbit, travelling right around the earth to hit you in the back of the head 90 minutes later.
So, what would we have to do to get a payload off the earth, and to the sun? First we would have to get it away from the earth, and that is not easy. But once it is away from the earth, it would still be orbiting the Sun. We would need to slow it down, almost to a stop - from 30km/sec down to zero. That is hard. Remember, there is no friction out there, no tyres on a road. You have to use a rocket engine pushing backwards, and it is just as hard to slow down in space as it is to speed up. The measurement of a space craft's ability to change speed is an important number, and is measured in meters per second - we call it the 'delta-V' of a spacecraft. The entire Saturn-V moon rocket had a delta-V of about 18 kilometers per second - without a payload. So magic a complete Saturn-V rocket away from earth, and it could get a tiny payload about two thirds of the way there.
Of course, once you did stop the rock, you would not need to push it toward the Sun. It would just fall - like a stone - towards the sun, reaching a speed of hundreds of kilometres per second before burning up from its own velocity. The Sun's heat would be nothing to the rock's own energy.
By the way, that Messenger probe to Mercury is a good example of how hard it is to get something to go towards the sun. In order to reach and place itself in orbit around mercury, the spacecraft was send on a multi-year path that took it once past Earth, twice past Venus, and three times past Mercury.
Well, this was much longer than I thought it would be!