Mac – How to use the Time Machine command line interface to backup to an internal ip server

Both are feasible. I use the first myself, and my sister had the second set up at one point.

1) On the laptop, you can install netatalk which handles AFP file sharing. Time Machine requires some special AFP features but those are supported by the version of netatalk that comes with Ubuntu 12.04 (just do sudo apt-get install netatalk).

Here's a decent guide for that, although with current versions you shouldn't actually need avahi-daemon, netatalk can handle Bonjour by itself now. You may actually just be able to get away with adding a line like:

/home/username/backup    "Time Machine"  allow:username   options:tm,noadouble

to /etc/init.d/netatalk/AppleVolumes.default, which is all I had to do on one machine running 12.04 (I don't remember having to do the other stuff, but YMMV).

Note that you should NOT ever need to use the "EnableUnsupportedVolumes" hack as netatalk meets Apple's spec and should appear as a backup destination automatically. Using this hack causes Time Machine to ignore problems that can cause loss of your backups.

2) A Time Capsule can indeed be connected to a wireless network as a client to share the hard drive (it can also share a printer connected via USB). I think the setup wizard makes it pretty easy, you're basically just looking to set it up as a wireless client and not a router. You might want to hook it up by ethernet if the signal isn't so great, though.

If 320GB is enough space (which I imagine it is, unless your Air has a 512GB SSD), I'd go with the netatalk solution unless money is no object.

Yes. OS X Server specifically includes the Time Machine Server software that makes it easier for Time Machine backups from machines on your network to reach, wakeup and perform their Time Machine backups to the server using native Time Machine data transfer protocols.

OS X Server can act as a designated Time Machine backup location for all the Mac computers on your network. Centralizing your backup storage helps protect valuable data and free up disk space on individual drives. And it eliminates the need for separate backup drives altogether. OS X Server also takes advantage of Time Machine to back up your server data — including shared files, calendars, mail, wikis, and more — to another hard drive, so you can easily restore to an earlier configuration. And you can view the backup status for each Mac on your network, as well as which computers and users are backing up to your server, when they last backed up, and the total size of each backup.

It is the only officially supported way for performing Mac-to-Mac Time Machine backups by Apple.