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The phone has to run at somewhat lower than 3.7 V, the nominal voltage for those batteries, so use 2 diodes in series.
The voltage drop across a diode depends logarithmically on the current through it, so if the phone current changes much (such as when transmitting), this may not work well.
I thought the voltage drop across a diode is always the same no matter what.
Google "Shockley diode equation".
I don't know if the capacitor trick would work, but many phone batteries have a thermistor terminal as well, and the phone might expect to see a certain range of resistance values.
Can't you just put the battery back in the phone? Then provide the 5v.
BTW: your title for this thread should give a clue as to what area your problem is in, not just "please help me". People on this forum like to help, but may not have time to read ever "please help me" thread.
You could....
Back the 5v on the first dc-dc converter off to 4.6v or something, and then do the diode trick
Use an appropriate LDO regulator to supply it.
Connect power to the charging connector of the phone - but you say you don't have the old battery, and it refuses to work without one
Back the 5v on the first dc-dc converter off to like 4v and call it good (but then you need to disconnect the phone when the Arduino is plugged into USB, since that would give it 4.7) - the '328p (as used in nano/pro mini/uno/etc) is still in spec at 4v, you'll have to check the specs for any other devices you've got connected to it
Ardunaut:
What does it mean ? If to current increases, the voltage drop gets bigger ? I thought the voltage drop across a diode is always the same no matter what.
You need to look a datasheet. Diodes are not voltage sources, or anything like, especially
at high currents where the series resistance dominates.
Since you are powering a phone you'll need high peak currents (much higher than you
think I suspect). There is no substitute for a proper high current supply such as
a 2A DC-DC converter (something like that level). Drive this directly from your input
voltage so it doesn't load the 5V rail or you'll pick up lots of pulsing on the Arduino
supply when the phone is transmitting packets.
Extra decoupling on the phone supply is probably a good idea too, as much as 1000uF
wouldn't go amiss.