I am a 1st class Sanskrit (H) graduate &1st class in Masters (history). Am I eligible for NET exam?

I don't not understand  language 

No, UGC NET is not a mandatory entrance exam for NIT Srinagar PhD admissions. Candidates can also appear in the GATE as well to qualify for admissions to the NIT Srinagar PhD course.

However, candidates who are applying through a ME/MTech score are required to appear for the national level NIT Srinag

No, candidates who have either appeared for the GATE or UGC NET are not required to appear for the NIT Jalandhar PhD admission Written Test. The aforementioned written test is only applicable to part-time PhD admissions where the candidates have not qualified in either the GATE or UGC NET.

The NTA will release the UGC NET 2026 June City advance city intimation slip around 10 days prior to the exam date. It will be released at ugcnet.nta.nic.in. 
UGC NET intimation slip mentions the date of examination, shift, subject/test papers and medium chosen by the candidate while filling the

The National Testing Agency is expected to conduct the June session UGC NET exam 2026 in the last week of June. The whole schedule will be announced on the official portal at ugcnet.nta.nic.in. The notification is likely to be announced in the April 2026.