Looking at the OGR API itself, I think you need to go via feature.GetDefnRef, to get the feature definition, then .GetFieldDefn for the relevant field, and finally .GetNameRef...?
...
feature.get_field_count.times do |field_index|
defn_ref = feature.get_defn_ref
field_defn = defn_ref.get_field_defn(field_index)
field_name = field_defn.get_name
field_value = feature.get_field(field_index)
puts "Value = #{field_value} for field named #{field_name}"
end
...