Slowing down the Earth's rotation all at once would be a catastrophic change no matter how it's accomplished. Everything on the Earth's surface (including the surface itself) has a characteristic angular velocity as a result of the Earth's spin; my back-of-the-envelope math says it's in the vicinity of 1600 kph at the equator (it varies with latitude and also slightly with elevation).
You're going to be increasing the Earth's rotation period by a factor of 365,000 (from one day to 1000 years), which means the rotation speed is reduced by the same factor. The 1/365,000th that remains is basically zero. If this happens instantaneously, everything on Earth will go flying off at its current angular velocity. Since this is less than escape velocity, it will eventually plummet back to Earth. Neither of these does any favors to squishy things like humans and infrastructure.
Even if the change takes place over a period of time, there's the potential for damage. If we allow for one tenth the force of gravity (0.1g or .98 meters per second per second) at the equator, we can slow the rotation of the Earth safely in about 186 years, assuming the force remains constant. I don't know what the actual safe margin for e.g. buildings would be; I assume it's not all that much, since unexpected ongoing sideways acceleration isn't generally planned for in architecture. You might want to build in a little extra buffer to avoid damage. Conversely, if you decide you don't like the equator that much, it might be possible to push that limit without hurting the higher latitudes too badly.
This will wreak havoc with geostationary satellites; they won't deorbit or anything, but they also won't be geostationary and so they won't work right. Given time, we could probably replace them with less efficient satellites in new orbits. And, of course, changing the day-night cycle will eventually cause severe disruption in animals, plants, and people. (Although it may be gradual enough for humans to cope, and some animals, many animals' whole survival strategy will be invalidated and they'll die off. Plants might hang on for awhile, but won't be able to handle 500-year-long nights without drastic changes.)
Being pulled out of orbit isn't too difficult. Earth's orbit is the result of the interplay of its orbital velocity and the Sun's gravity. Adding more velocity will result in a larger orbit, neat as you please. You should be well within tolerances adjusting the Earth's orbit in your mission timeframe. As you might expect, the further the Earth is from the Sun, the less sunlight it receives and so the colder it becomes, and the less energy there is for plants and solar power.
However, there is a problem. A single pass by an object obviously won't produce the small force over the course of centuries that would make this survivable. An object could impart that kind of force if it was captured and made a (semi-)permanent part of the Earth-Moon system. However, if the object is large enough to affect the Earth's own orbit, it will destabilize this system, which could be catastrophic.
If you're okay with having an obviously artificial cause for this calamity, you could posit an alien (or advanced human, if for some reason this struck humanity as a good idea) probe that orbits near Earth, using a combination of gravity tractoring and ion-beam shepherding (using, respectively, its gravity and its propulsion exhaust to make minute changes in the Earth's orbital characteristics over a very long period). The engineering requirements would be on the extreme side but it's theoretically sound, and the probe being powered lets you skip a lot of complications about how it ends up in a stable orbit.
Incidentally, I'm sidestepping the idea of changing the Sun's mass because a) there isn't really a mechanism for that happening naturally, to any great extent, b) weird things happen to stars when you fiddle with their masses (for instance, removing a chunk of the Sun would reduce the luminosity of the rest), and c) that would only affect Earth's orbit, not its rotation.