No of course not. Manufactuers always make the recommended wattage high.
I just would not be comfortable using an old 240w unit with my new parts in it.
The optiplex 390 is quite an old pc and most likely has been used quite alot so i wouldnt compare the psu he currently has to a brand new dell psu.. The 12V rail wont hold up as well as the unit ages.
Is it possible to run a 1650 on a 240w psu? Yes.
Is it recommended? No.
it depend where its been used..usually optiplex are office pc, they are mostly 20% load on usage for checking email, microsoft office. Believe it or not, those PSU actually last more than "gaming" corsair/thermaltake/evga psu
if the psu is dead that would be the time he would need an upgrade..or he could fix it, its either fuse or bad capacitor. but most of the time its fuse on older optiplex psu. a lot of people dont bother doing fixing their dead psu as it take too much time doing solder. but i rather spend 1$ on fuse on local store and solder it, than buying a new psu that would cost 35$. save money. older psu uses glass fuse, while new one uses poly fuse
to prevent repair.. older psu are easier to repair than new psu, they want to become the next iphone, when its broken buy new one.