A new LCD was probably unnecessary. Your Sony monitor should be good enough to not strain your eyes. You can do a few things to ensure the best picture:
1. Raise contrast to about 90%, then adjust the brightness to around 50-75%. Excessive brightness will lead to eye strain and faster CRT wear, so aim for closer to 50%. Obviously different CRTs will have different characteristics, but the various aperture grille monitors I've owned all looked fine with their brightness somewhere between 50 and 65%.
2. Lower the color temp to a *much* more comfortable (IMO) 6550K (the NTSC standard, BTW). Whites may look less fluorescent initially, but you'll get used to the slightly more reddish whites quickly, and they should be easier on your eyes. This made for a noticable comfort improvement with my two Mitsu aperture grilles.
3. Make sure you've got a video card with good signal quality driving your monitor. I'd recommend an ATi Radeon or any Matrox for your CRT. Earlier nVidia cards (GF1, 2, 2MX, 3, and even 4) weren't that great above 1024x768. This isn't true for all nVidia cards, of course, but from my rather obsessive forum-watching, I feel it's safe to say ATi cards have better signal quality at higher resolutions, something you'll appreciate with a nice monitor like your Sony. If you're running that beautiful monitor off an integrated graphics card (built into the motherboard), run, don't walk, to buy a discrete card. Ask if you want a more specific recommendation--just specify if you're a gamer or not (and what games you play, if you are).
4. Make sure you're running a comfortable refresh rate of 85-100Hz.
5. Make sure you're running a comfortable res, either 1152x864 or 1280x960 for a 19".
6. Don't forget to occasionally blink, drink water, and focus on something 10-15ft away, especially if you're staring at the monitor for a few hours a day.
Anyway, everything applies to a new LCD, except that you'll want to make sure you're driving it via your video card's DVI connector.
Edit: 120+Hz seems excessive, both for video cards and for CRTs. I'm comfortable with 85-100Hz, but I guess it can't hurt to try 120Hz. I do recall reading that that high a refresh rate won't do your monitor's lifespan any favors, though I may be repeating nonsense.