If I understand your problem correctly, you made that transaction.

0.001   BTC went to 1Jxw*
0.045+ BTC went to 1Cb5*
0.0001 BTC was the fee

So 0.0011 is the amount you've sent out + the fee (total)
And I expect that the missing 0.045+ BTC should be in a change address in your Bitcoin core wallet (if you didn't remove/change/overwrite the correct file).

What I would do:

1. Make a copy of wallet.dat from you data dir ( see https://en.bitcoin.it/wiki/Data_directory ) into a safe place as backup.
Make sure that Bitcoin Core is fully stopped when you do that.
Make sure you understand what safe copy means (it doesn't mean dropbox, gmail or such; another place on your HDD + one more copy onto usb stick would be nice)

2. Sync your BitcoinCore if you have enough HDD space for it (300 GB) and time. It's the easiest.
Alternatively, if you don't have the time you'll need guidance about pruning or about finding out the private key of 1Cb5* and importing that into a SPV (light) wallet like Electrum.