I can't tell you how to overcome your specific "Eee PC issue", i.e. having neither a recovery disc nor a CD drive, but I can tell you that "FIXMBR" is definitely the right way to solve your problem - it just writes a new MBR to your internal HD and that's what you want. You won't encounter "another error" here when done right - I did this countless times already without any hassles afterwards.I have read somewhere that one could use XP recovery disc run repair fixmbr, but I also saw someone following that direction only to encounter another error.
As of your specific problem: Booting XP from an USB flash media is possible assumed you have an installation disc to prepare the stick from. That installation disc does not urgently need to be the recovery disc of your particular Eee PC though (that needs to be enough as a hint here), so I'd try to get an installation disc and watch out for possibilities to create a bootable USB flash media out of it. Maybe there's even another method to recover the MBR but not to my knowledge at least.