Are you allowed to Daven there?
The Sefer Gesher Hachayim (29: sk11) writes:
"R. Shmuel Salant Zt"l would only Daven in the outer room of Kever Rochel* and not in the inner room.
He held that there is an Issur of לועג לרש (mocking the poor)
Davening in the inside room."
There is a Halacha Mefureshes in Shulchan Y.D. 367- 3:
" One may not Daven within 4 Amos of a Niftar."
The Shu"t Minchas Elozor 3-53 says that Kever Rochel is an exception for various reasons and defends those Davening inside the room.
His Heter is only applicable for Kever Rochel & not for other
Kevarim even Kivrei Tzadikim.
Today this Halacha in Shulchan is almost totally ignored. There
are very few who know of or keep this Halacha and don't Daven
or say Tehillim unless they are distanced from the Kever 4 Amos.
There is a Sefer called מענה הלשון which has special תחינות
(Supplications) to be said at the Kever. It was composed
for this reason so that one can come close to the Kever
and say these תחינות instead of Tehillim etc.
*The מערת המכפלה,ציון הרשב"י etc. are not problematic since they
are buried down in the מערה and the Davening is done on top.