Question: suppose the machine is clean of all malware but not in any sense updated, patched, secured etc. Suppose I connect it to the internet from behind wireless router with the intent of using it only on a few trusted sites and only there. Or, for the sake of argument, maybe I wouldn't do any browsing at all, just let it sit there connected to the network. This is all happening in a residential situation with cable internet.

In this situation can a remote attacker somehow detect the fact that the machine is connected to the internet and try connecting to do an exploit?

Answer: 
The second "clean" machine cannot be directly attacked and infected from the internet if it is behind a router but if your first machine has a known malware infection then it is possible that the malware on it could be written to actively seek out other machines on your network and infect them by any means possible.

If there is one machine on your network that is infected then all your machines are potentially at risk, especially if they share data or programs or usernames and passwords.

If this clean machine also has an older or unpatched operating system it makes it more likely that it has vulnerabilities that could be exploited across a home network.

If you only ever go to absolutely trusted sites then you may well be okay but the first site I would go to would be an antivirus site to get some up to date protection.

Until you can clean the infected machine I would only ever have one machine switched on at any given time.
Whether a computer can be discovered behind a router has nothing to do with whether it is up to date and properly patched, but whether the router will allow it to be accessed. The [network address translation](http://en.wikipedia.org/wiki/Network_address_translation) and firewall provided by the router may provide a (small) degree of protection against the system, but it is still relatively easy to detect and exploit the vulnerable computer.