Power old Nokia phone from Bench Power Supply
Is that a Nokia 6310i (or one of it's re-branded brethren)?

If so, then I believe the answer is no.  It is not possible to power this phone up using only a bench PSU.  You would have to provide some electronics to trick the phone into thinking there was a real battery present.

I have just tried mine on a benchtop PSU (connecting +3.6 V and GND only, 200 mA current limit) and it refuses to power up.  It momentarily draws about 130 mA when I press the power button but, as has been said in the comments above, it is likely that the battery needs to be proven to be "healthy" and responsive before the phone will allow itself to be used.

I put my battery back in and it works fine.  It only has the 4 connections at the top.  So you can ignore the ones at the bottom for your purposes.

For your information the battery bears the following information (verbatim)...

RECHARGEABLE Li-Polymer

Type BPS-2 3.6V

BL0893213E

My battery still lasts a long time (2 weeks of standby time from a full charge) and it's probably 15 years old.  It's likely that you can just buy a battery and charger second hand.

Ahhh, they don't make 'em like that any more! :)