Can I make a switched outlet which is connected to light hot all the time without running new wire?
The only hot leads going to the outlet box are switched (both fan and light black wires). Since there is no unswitched hot lead, you cannot have constant power to the outlet without running new wires.

If you could tolerate one switch controlling both fan and light, you could create a constant hot at the outlet by bypassing either the fan or light switch and connecting the two switch wires together. Then one of the blacks at the outlet box would be constant hot. That would be your hot lead to the outlet, rather than continuing up to the fixture. You would cap the unused wire to the fixture on both ends. Finally you would connect both the fan and light hot leads to the one remaining switched hot. But, obviously you lose independent control of the two functions.