If I invent something that solves a problem in a new area (an
improvement to something that was only invented ten years ago), by
using a technique and mechanism disclosed in old invention (something
that is 50 years old, in fact), and that no one else has thought of
applying to the new area, is this patentable
Finding new uses for existing technologies happens all the time and is potentially patentable. The original inventors of the laser would have had no idea of its use in flow cytometry.
or will the examiner say "It is obvious to apply the old invention to
the new invention" even though no one has done so in the last decade?
Yes this is a definite possibility. I am not a lawyer so strategies to avoid an obviousness objection are beyond my skill set. That said, I think you are safer if the application of the existing technology in the new application comes from an unexpected benefit. So, for instance, applying a teflon coating to a new device in an effort to reduce friction is likely to be rejected as obvious. Applying teflon to to a new device to improve radio frequency transmission might be OK.
If the examiner does take this line, are there any persuasive
arguments or former cases that could be drawn on to overcome such an
objection?
I have to leave the answer to this question to the actual lawyers.