The letter has no apparent letterhead, so I am not sure if it is legitimate, although the text follows pretty closely what is known about how Lebanon is treating its Syrian refugees of Palestinian ancestry.
Here's the translation:
The Syrian Arab Republic
Ministry of Interior
Passport and Immigration Department
The Palestinian Emigration Department
The cases in which it is allowed to travel to Lebanon, knowing that the prohibition is by the Lebanese authorities alone
1. Those who have valid residence permits in Lebanon
2. Those students who study in Lebanon and have student cards for this year or documents proving that they are registered in Lebanese schools
3. Palestinians whose mothers are Lebanese or Palestinian Lebanese
4. Those traveling through Lebanon and have a visa and a plane ticket, 10 hours before the time of the flight
5. Other cases must be approved by the Lebanese Embassy in Syria
This fits in with the recent HRW press release on the topic as well as various reports over the past couple of years.
What is clearly true is that no Arab nation is treating Palestinian refugees from Syria the same as they treat non-Palestinian Syrian refugees. (I am not aware of whether Turkey differentiates between the two groups.)