A lot of modern serif typefaces actually date back to that era, but to get the appearance of an antique newspaper, you need to simulate how the ink behaves on the paper. First of all, it tends to splodge out a bit, making fine details a lot thicker than they would otherwise be. You also tend to see a faint ghost of the text printed on the other side of the paper caused by the ink soaking through. The first of these effects can easily be simulated in CSS by adding a
-webkit-text-stroke-width rule. (You might need to increase the letter spacing a bit at the same time in order to preserve legibility.) The second can be simulated by using a seamless background made of blurred text that has been reflected horizontally.
Other things you can try include using old-fashioned typographical tricks like small caps and/or drop capitals for the first line of an article. Since newpaper articles are usually printed in narrow columns with full justification, you should probably avoid long lines of text, and include plenty of soft hyphens (
­) to prevent large gaps from appearing.
Here's an example I made using the Baskerville typeface (which has a free alternative available at Google Fonts). You can play around with it here.
In case you need it, here's the background image I'm using. It's just made with Lipsum set in a similar font: