My experience when a rim is advertised with an inches width [Rather than an internal diameter in MM] it's much more of a general statement. Chances are that you will be able to use the exact same tire you used before.
Generally a 1.75" tire fits just fine on rims with internal widths of 17mm to 26mm with out any real hazard. If you go outside of that range there is a higher chance of burping the tire, having a blowout, or popping a bead on hard turning. A 2.00" tire will safely fit on a 19mm to 28mm internal width. So if the inner width of your new rim falls between 19 and 26mm you can use the exact same tire you had previously. If the internal width is 17 or 18mm then i would consider a slightly narrower tire such as the 1.75" or a 1.95"
With that said i would check to see if they list an internal rim width on the site your looking at, this will be an indicator as to what sizes of tire it will work best with.
Having a more narrow tire on the front however is neither uncommon nor problematic.