How do ARM able to compete with old x86 cpu's like athlonxp 1800+ in single threaded applications?
They're not meant to be compared, they're used largely for different purposes (ARM tends to be implemented in mobile devices and is more about power saving them performance).
An ARM based aplication cannot run on a x86 cpu and vice versa, but if you compare raw performance, like instructions per second, memory speed etc, modern ARM cpus perform (obiously) better than old x86 chips.