I'm going to say yes, but I forget how to do it - I seem to recall seeing details in the handbook but it's out in the car and it's raining so it'll have to wait til the morning
You should post a link to the source of that PDF -- it's amazing how many of the questions asked on this forum can be answered by simply referring to the manual
Btw, just FYI, you can get any manual for Fiat car if you register in "My Fiat" on fiat.co.uk, and then log in and click on faint grey "My Fiat" on the bottom of the site.
Not sure what battery it is but once you find out which it is have a look in your local Pound Stretcher they do packs for £1 for 8 of the "big" ones or 30 "little ones" there will be a code on it like "2092" or "AG3" and thats what your looking for
The answer to 'what's the battery' can be found by simply opening the key (like I just have) -- its written in the battery. Its CR 2032 -- slightly smaller and thinner than a 10p piece.
As to needing to have a spare in case it goes flat: firstly there's a low battery warning (the LED only flashing once when you press the buttons instead of several flashes if you keep holding it)... and secondly, if it doesn't work, you simply put the key in the lock of the driver's door and turn it -- as described on page 10 of the instruction book from Ingvaar's helpful link , where it says:
"IMPORTANT If the remote control does not work properly, it is still possible to carry out the above mentioned operations [those being locking and unlocking the car] by using the metal insert of the key."