Note: I haven't included communication with other humans in my answer, I'm assuming the ship being at the Oort cloud cannot communicate with other ships.
Context
It's possible for the human brain to solve these problems quite easily, as can be seen by playing Kerbal Space Program and the Apollo moon landing was achieved with roughly the computation power of a calculator.
Scanning
The question is, what does the computer need to know. Assuming it has a radar that can output 3D vectors of solid objects you then to store these vectors. You need to decide some resolution and range for the radar. The distance to the moon is 384,000km so we'll make sure the range is big enough to encompass that, let's say 400,000km. (I'm making the assumption here that you don't want your radar to be able to track as far as the Oort cloud, you just want to be able to process reactions to collisions.)
We'll want to be able to pick up fairly small objects on this radar and avoid them so we can split that into units, being practical the 67P/Churyumov–Gerasimenko comet we just landed on is about 4km accross, so we'll want to pick up on that. That means at maximum that we have 100,000^3 units of scanning data to store.
Assuming the positions are measured as integers that's 1.2x10^16 bits of data, or 1.5x10^15 bytes (That's 1500 Terrabytes!) if the entire field of view is covered in matter. The ship is probably not going to be surrounded by objects so we can give it some kind of buffer and say that it can store maybe 100 points, which is of course a lot less, about 150 bytes. Which is really easy to handle.
Control
That's just for the scanning data. My calculator can only store about 70 numbers which means it probably can't do what you want. Lets assume a minimal computer with 1Kb of RAM which is 1024 bytes. That's a bit more than we need so it leaves us with a healthy 874 bytes to play with.
What do we need to do with this? Here you need to consider the structure of the ship. I'm going to go with the simplest possible configuration (based on Kerbal Space Program missions):
* 1 Reactionary engine (3 degrees of rotation freedom!)
* 1 Thruster (1 degree of forward momentum!)
[This just includes motion stuff, I'd expect door, lights, air locks, life support etc to all be simply plugged into some power supply and left to run, turned on and off light light switches.]
We need to be able to control these and make calculations about how to move the ship. 199
So we need to know a few things:
* Velocity (3D vector)
* Acceleration (3D vector)
* Rotation (Probably a Quaternion, 4D Vector)
These need to be pretty damn precise so lets make them big numbers and say that we need 160 bytes for these! So we're left with 714 bytes to play with.
Now we need to do some maths. Assuming the ship knows its own mass it probably only needs to do $F = ma$ and $F = G({m_1 m_2}/{r^2})$ a lot to avoid an object. It might also want to then store a few values about its positions after a move its going to make, validate whether or not that leads to crashing into another different object and try again. So we can add another 100 position values or 150 bytes bring us to, 564 bytes left.
These remaining 500ish bytes of RAM must simply contain some basic logic about the process I described above.
1. Look ahead some amount and predict a crash
2. Try a move and see where it takes the ship
3. If that's no good try another maybe in the other direction
4. Try and correct movement towards some target vector if the ship has a destination
All of this can achieved with very little memory.
Processor
I don't know a lot about processors, but these calculations use very small volumes of data. My calculator lags when I press the button and we don't want that, so something in the range of 100+Mhz would probably be okay.
Answer
This all assumes that the calculations are this simple, I've literally stripped the system to its bare minimum. You could say that a few Mb of RAM would be enough for a simple purpose computer and a processor that runs at about 1Ghz. This would not be fast by our current day standards as my phone has 1Gb of RAM and 2.4Ghz clock rate for its processor. Adding more processing would allow for better gravitational calculations and collision prediction. 234