Opened up a USB stick and it looks a bit odd

TestDisk is a tool for file and partition recovery from damaged/deleted/corrupt partitions. If that doesn't detect it, then it's unlikely you'll be able to get anything off. (If your running under Linux make sure you run it with sudo)

If like you say it's not getting power, then I would guess that something has failed internally. If the data is important take it to a specialist.

You can do what you want very easily.

Assumming only a two partition setup, you have to make the FAT partition your first primary partition (it has to be in the same order in the partition table and on the disk, it's usually the case, but strange things happen in Windows if you don't).

On your EXT partition install Ubuntu.

On the MBR install grub as following (I assume you already have a linux machine and grub2 working on it and that your pendrive is detected as /dev/sdb):

mount -t ext2 /dev/sdb2 /mnt
grub-install  --no-floppy --root-directory=/mnt/ /dev/sdb

This will setup grub on mbr and install all necessary files on your ubuntu partition. Next you need to setup your grub.cfg with a menuentry as follows (read grub2 documentation or the grub.cfg on your linux pc to build a full grub.cfg):

menuentry 'Ubuntu, con Linux 3.2.0-39-generic-pae' --class ubuntu --class gnu-linux --class gnu --class os {
        insmod gzio
        insmod part_msdos
        insmod ext2
        set root='(hd1,msdos2)'
        search --no-floppy --fs-uuid --set=root UBUNTU_PARTITION_UUID
        linux   /boot/vmlinuz-3.2.0-39-generic-pae root=UBUNTU_PARTITION_UUID ro   quiet nosplash
        initrd  /boot/initrd.img-3.2.0-39-generic-pae
}

You need your partition UUID for this to work on computer with more than one disk:

# blkid /dev/sdb2
/dev/sdb2: LABEL="GPART" UUID="75cdfe28-0ad8-4f0a-8c1d-9af6b2a5ba88" TYPE="ext2"

I'll leave it up to you to find out how to add a second entry for memtest86+, which is perfect for this kind of rescue flash drive.