It might work. The regulator on the Arduino is recommended for 7-12V, but the actual limits are 6-20V (See [u]this page[/u].)
I think you'll be OK. However if you are powering the LCD and clock from the Arduino's regulator, you might be stressing it. (The more voltage you "drop" across the regulator and the more current through it, the hotter it gets.)
Another option is to use a [u]7809 Voltage Regulator[/u] (and a couple of capacitors) to knock the 18V down to 9V. That way, your external (bigger) regulator is doing most of the work.