Yes, you can use genome walking kits. But that might be expensive. Alternatively you can use 5' and 3' RACE. That will give you the full sequence from your partial sequence.
You can find homologous sequences of the gene if you know the gene. If not, you can blast the known sequence and find out homologous sequences from other species. Then design few sets of primers in the region that has highest homology. If you get a good PCR product, then you can send that for sequencing. You can use cDNA as the starting template if possible.
Then I assume the sequence is not deposited. Even a homolog sequence may help to design a reverse oligo (guessmer). An alternative is to cut the genomic sequence with a restriction enzymes, in your case e. g. Sau3A. Then do a religation. In this step the DNA should not be too concentrated. Design 2 Oligos in opposite direction, do a PCR and sequence the PCR product. The position of the unique Sau3A site allows do define 5' and 3' flanking sequences. Since there is the risk that a Sau3A site is too close, you may perform the procedure with a second enzyme.