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.

Conventionally, no

Conventionally, your lamp absolutely requires a switched-hot and a neutral wire, and that uses up all the wires you have available. You are at impasse and cannot proceed.

Your only option would be to put the light switches at the lamp via pull-cords and the like (though, building codes may have a big problem with this) and reassign the two available wires to always-hot and neutral.

Unconventionally, yes

This can be done with smart switches. We are not a shopping site, so it isn't for us to recommend brands. But the general idea is that you re-task your two wires to be always-hot and neutral, so that all sites have power all the time.

In the switch location goes a smart switch which is able to transmit signals either over the powerline, or via radio (wireless). Inside the lamp, underneath the ceiling rose, goes a partner electronic module to the smart switch. It inputs always-hot and neutral, and listens to the powerline/radio for a "turn on/turn off" command sent from the smart switch. It outputs "switched-hot", which goes the few inches to the lamp proper. The lamp also uses the neutral which is present.

GFCI is a snag

One problem with powerline-transmitting smart switches is they cannot transmit through a GFCI device. You may be tempted to slap up a GFCI+receptacle combo device at the outlet location to satisfy Code and for the genuine safety value it provides. If you do, don't use the LOAD terminals. Pigtail the downward wires and attach only to the LINE terminals. I recommend this very often, but in this case it's to assure the light and switch are on the same side of the GFCI device.

Now if you have a GFCI device (such as a GFCI deadfront or breaker) before the switch that protects this circuit, then the whole works is already on the load side of that GFCI, and switch and light should be able to communicate just fine.