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Have you considered using a couple of hard disc drive bearings to support the shaft?
I have been searching for watchmakers lathe attachments for several years now and I have never seen a drilling attachment for the tool post. The usual method is to use a vertical slide with a spindle that can be set at any angle and takes WW collets.
Consider getting a 8mm watchmakers lathe headstock as a beginning point and make your own.
If you search ebay for drill chuck motor mount you should find a bunch of small 0-4mm chucks with a JT0 mount. Some come with motors and some come with the chuck and motor mount. I see mounts for 5mm shafts and 1/8" shafts. That should solve the problem of not having a center drill.