The Wii Shop channel is still up. Since you have an older Wii, try doing a system update and making sure that you are connected to the internet.

There is no official way to apply updates to a Wii apart from connecting it to the Internet.

Wii games are never patched: Nintendo has a strict no-patch policy, with the idea that a game shouldn't ever need patches after release to be feature-complete or free of major bugs. The result of this is threefold:

• You don't need to connect it to the internet to play games that don't have DLC features
• Games that are obviously unplayable at release just don't happen
• The rare major bug that does slip through QA is a big embarrassment for Nintento and the publisher

The Wii does need a network connection for some non-essential features. These include:

• Game DLC, leaderboards, and multiplayer
• Wii messaging, polls, news, weather, Mii Parade, and other non-game system features
• Buying and downloading Virtual Console games

The one essential thing the Wii needs the network for is when a newly-released game requires a particular system version in order to run. The first game I noticed this with was Rock Band: Beatles, which wouldn't run until I updated to (if I recall correctly) system menu 4.3. With a newly-purchased Wii you won't run into this for some time, but when you eventually do it will be a show-stopper if the Wii doesn't have network access.

Even for system updates and features that do require a network connection, a constant connection is not required. Once the data is on the Wii or the update applied, you can disconnect the Wii from the network and use the Wii normally.