Is it ever possible or permissable for a Protestant to go to Confession? I know I will not receive absolution from the priest like a Catholic would, but I do feel the need to confess.
It is possible. Of course, to validly receive a sacrament other than baptism, a person must be baptized (c. 842 §1.)
A baptized Protestant might lawfully receive the sacrament of penance from a Catholic priest under the conditions of Canon 844 §4.
“If the danger of death is present or if, in the judgment of the diocesan bishop or conference of bishops, some other grave necessity urges it, Catholic ministers administer these same sacraments (sacraments of penance, Eucharist, and anointing of the sick ) licitly also to other Christians not having full communion with the Catholic Church, who cannot approach a minister of their own community and who seek such on their own accord, provided that they manifest Catholic faith in respect to these sacraments and are properly disposed.”
Since Protestant denominations have no sacramental absolution, that particular condition of approaching one’s own minister seems extraneous to their situation. However, the priest should know if the other conditions apply in your case, and if they do, can lawfully give absolution as well.