I do not own the game, but have played it numerous times before with friends...good game overall...
1) Yes, it does have an online play feature. It is actually the first Tiger Woods golf title for Wii to feature online competition. Make sure your Wii is successfully connected to the internet. Then, select the "online" option on the main menu. Follow the prompts to connect, and you can play on EA Sports' online network.
2) However - This game was first released back on August 26, 2008. You may not find many people online playing this version any longer, as competitive online golf tends to lean toward the "latest and greatest," AND a new TW Golf has come out annually. Many online-golfers in this game segment tend to upgrade annually, and "fore-go" (pardon the pun) the older versions. Don't get me wrong - you will still find people playing it, but not as many as other "dated" titles online.
Best of luck! And no - you may NOT pick up that last 4 footer for par. Putt it out like a man!