Yes. 4GB of RAM at the minimum. If you are near an Apple store or Apple certified repair provider, you can have them install RAM for you.
If not, if you feel confident about doing this yourself, you can purchase RAM online.
Crucial is highly recommeneded. You can use their scanner tool > SDRAM, DDR2 and DDR RAM memory upgrades from Crucial.com
And before upgrading to Mountain Lion > App Compatibility Table - RoaringApps - App compatibility and feature support for OS X & iOS
Be aware, PowerPC apps that require Rosetta cannot run on Mountain Lion.
For downloading Mountain Lion from the App Store, a high speed (broadband) internet connection is strongly recommended by Apple as noted here > iTUNES STORE - MAC APP STORE - TERMS AND CONDITIONS
If you need to reinstall OS X or repair the the startup disk using Mountain Lion Recovery, that requires broadband (high speed) access to the internet via Wi-Fi or an Ethernet connection. OS X is downloaded over the internet form Apple when OS X Recovery is used for reinstallation.
As far as your Mac running slower after upgrading, make sure there's enough available space on the startup disk.
Right or control click your MacintoshHD icon on your Desktop then click Get Info. You'll see Capacity & Available.
Make sure there's at least 15% available disk space. Anything less may slow your Mac down.