If I have the reverse transcription products from total cellular RNA, is it possible to sequence fragments with specific primers without gel purification of the cdna first?
Directly you may not do sequencing, because while sequencing you are supposed to use single primer. If the product is not purified, it may contain multiple templates which may get amplified possibly with a single primer and may produce a lot of noise which could reduce the quality of read and even it some can hinder amplification of your target.
If you use only one primer per reaction and reverse transcriptase, it should be theoretically possible, but reverse-transcription to cDNA first would be better ;)
First I would recommend you to do a normal PCR with the primers and check the amplified product in a gel. If it is having only a sharp single band then you don't have to purify the product by gel extraction, but do use the PCR product for your sequencing reaction. Very often the primers, however specific it might be, tend to amplify multiple regions. So if you don't try your primers first then it might give you noise in your sequencing result.
I would recommend you to amplify the regions you want to sequence with a pair a of primers, just a normal PCR using cDNA as a template. If you get a nice&specific band you don't need to do gel extraction. There are other products (Illustra Exostar is the one we use) that clean your reaction enough to be used for sequencing. Gel extraction can be tricky, so if the PCR product is good no need to do that.