How does the paperweight represent the past?

The paperweight represents both the past and present for Winston and Julia. 

It is the past because nothing like it exists anymore.  It was produced in a time when things were not as they are with the Party.  It was a freer time, a more beautiful time.  It is an "antique".

Winston is drawn to it for these reasons.  He is obsessed with the past as it is something close to what he would like to have now--a life like that of the Proles, free to do what you want when you want.  To love freely and sing out loud and show emotion when you feel it.

It also represents the present for the couple.  Like the paperweight, their love is fragile and beautiful.  It is also forbidden.  Party members are not to show emotions or have relationships outside of their duties.  The coral center is the love they feel for one another, and like their affair, the clear glass that surrounds it represents the fact that Winston and Julia's affair was never hidden.  They were always loving inside a fish bowl...it was only a matter of time before they were caught and punished.

When the thought police finally raid their apartment and capture them, the paperweight is thrown carelessly aside and it smashes into a thousand little bits.  This is symbolic of both the past and their present, and it foreshadows their future.  They will also be smashed into bits and will live a fragmented life in the future--one devoid of each other.