Hello. I am using a very old guitar and the nut seems to be thinner. I changed the strings and the low e as well as the a string seems to be hitting the first fret at 0. There is an irritating metallic ringing when I pluck at 0.

Can I put some paper between the nut and the string or something?
That would probably dampen the sound.
I would reccomend taking off the strings, knocking out the nut and buying a new one. They always have precut ones on ebay and just file it to size and glue it in. Google it, and there's heaps of tutorials.
Just don't file too low, I know from experience.
You'll just end up in the same place you're already in.
Hope it works out
If you're not sure about replacing and filing a new nut, take it to a local shop and have them do it for you. If you do tackle that project yourself, the key is to file slowly and keep checking it. You can always file more away, but you can't put it back on
Current Gear:
2002 Gibson Les Paul Standard
'57 AVRI Fender Stratocaster
MIJ Fender Jaguar Special HH
Marshall JVM410
Vox AC15 C2
yes you can build up the nut with super glue and baking soda.

but first just check the nut heights by putting a bit of paper under the string.

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If you do the baking soda/super glue trick, you should tape off all around the nut so you don't get superglue on the neck. Also, you'll often be left with a slot that is too high, meaning it may need to be filed down afterwards, and you really need proper nut files to do so.
I'd recommend not doing it unless you're confident you can do it properly. A tech shouldn't charge much to fix it, and they'll know what they're doing.
Oh I fixed it already. I cut a small piece of my thin nylon pick and put it in the slot. Thanks for the advice tho..