Can we recycle garbage with the principles of mass spectrometry?
Yes it is possible, but is very expensive and would be orders of magnitude more costly than what people are willing to pay for recycled materials.  Let me give two data points to explain why it is expensive.


Some commercial material is manufactured by mass spectrometry.  The starting material isn't garbage, but reasonably chemically purified metals of various types.  The product(s) are isotopically separated metals.  Iron 57 ($\ce{^{57}Fe}$) is an isotope that is naturally present in terrestrial iron supplies at an abundance of about 2%.  It is useful to researchers that study iron materials using various spectroscopic techniques.  Pure iron 57 costs hundreds of dollars per milligram.
Mass spectrometry requires very high voltage, a very high vacuum, and large amounts of purified gases; additionally, the throughput of material separated by mass is very low.  I think that for commercial isotope purification, magnetic sector-type instruments are used.  I'm not too familiar with them, so let me go through some rough numbers for a different type of mass spectrometry: time-of-flight: Several thousand volts are applied to packets of ions which leads to their separation. Total ion currents are as high as say $10^9$ particles per second, which corresponds to about 0.1 nanoamperes or $10^{-15}$ moles per second.  So to purify one mole of iron would require $10^{15}$ seconds, or several million years of continuous operation of very expensive machinery!  Even if other types of mass spectrometry do better than time-of-flight by three orders of magnitude, it will still be cost-prohibitive.$%edit$