How to get name of variable via their value in class C#

Well you could use reflection:

public static string GetJurisdictionCode(string jurisdiction)
{
    return typeof(JurisdictionList)
        .GetTypeInfo()
        .DeclaredFields
        .Where(f => (string) f.GetValue(null) == jurisdiction)
        .FirstOrDefault()
        ?.Name;
}

But personally I'd suggest building a Dictionary<string, string> to get from code to name, and then either just using:

return codes.Where(kvp => kvp.Value == jurisdiction)
            .FirstOrDefault()
            ?.Name;

... or build up a second dictionary to go in the other direction. If you wanted the named constants as well, you could always build up both dictionaries by reflection, e.g.

public static class JurisdictionList
{
    public const string NSW = "New South Wales";
    public const string NT = "Northern Territory";
    public const string QLD = "Queensland";
    public const string SA = "South Australia";
    public const string TAS = "Tasmania";
    public const string VIC = "Victoria";
    public const string WA = "Western Australia";

    private static readonly Dictionary<string, string> CodeToName =
        typeof(JurisdictionList)
            .GetTypeInfo()
            .DeclaredFields
            .Where(f => f.FieldType == typeof(string))
            .ToDictionary(f => f.Name, f => f.GetValue(null));

    private static readonly Dictionary NameToCode =
        CodeToName.ToDictionary(kvp => kvp.Value, kvp => kvp.Key);

    // Methods using the dictionaries here
}

You should also consider having an enum, then mappings from the enum to names.