What Steve has said is very sound advice.
The are many ways to get rid of a tapered bore, and some of the purists will have you making or buying all sorts of wierd and wonderful contraptions to get this simple job done. But seeing that you are very limited in resources, I will try to explain the way I personally would go.
You don't say what the cylinder is made of, as that does have a bearing on what the lap should be made of.
You basically use a lap of the the next softest material in the pecking order downwards, so a steel cylinder would use say brass (or bronze), a brass or bronze cylinder, you would use ali, for ali, I have used a nylon rod before now and had great results.
So I have done a sketch at the bottom to show the method I am going to describe.
What you do, is turn a rod about twice the length of your bore, so that it only just fits into the larger end of the tapered bore, and make sure it is the same diameter all the way along.
Next, you file or shape a very slight taper on about 1/4 of the length so that it fits a little way into the smaller end of the bore.
Get yourself a bit of hardwood, plus some grinding medium. For starters, I would use some very fine valve grinding paste, or if you haven't any of that, some liquid metal polish.
Dab a bit of the grinding agent onto the rod, and working around the lap, rub all over with the stick. What you are doing is making a very fine but accurate file to dress out the hole, by embedding the cutting agent into the face of the lap.
Now what you do is turn on your lathe at the slowest speed possible with this lap in the chuck, and gradually feed the small bore end over the lap, and in a slightly twisting motion in and out to 'file' away very gently at the excess metal. Be very careful as the two parts can easily 'gall' together, so you have to be ready all the time to let go of the cylinder if it starts to spin with the lap.
Once you have got all the tapered portion into the smaller end, turn the cylinder around and come in from the large end and repeat the 'filing' process. Whilst doing all this, keep recharging the lap with cutting agent, as it breaks down under use.
Eventually, you will get to a stage where all the straight bit is thru the bore. What I personally would do then is make a new lap, but with no tapered end on it, and using a very fine cutting agent, such as a very fine metal or paint cutting polish (or even toothpaste), reintroduce it into the bore for final lapping and polishing.
You might need to make a new piston after you have staightened out the bore. It all depends on how accurate you want everything to be.
I have most probably forgotten a few things, so if anyone can fill in the gaps, please do so.
Bogs