Get previous/next item of a given item in a List<>

        List<int> listInts = new List<int>();
        listInts.AddRange(new int[] { 1, 3, 5, 7, 9, 13 });
        int index = listInts.IndexOf(3); //The index here would be "1"
        index++; //Check first if the index is in the length
        int element = listInts[index]; //element = 5

I have implemented this by Inheriting the .Net List

public class NavigationList<T> : List<T>
    {
        private int _currentIndex = 0;
        public int CurrentIndex
        {
            get
            {
                if (_currentIndex > Count - 1) { _currentIndex = Count - 1; }
                if (_currentIndex < 0) { _currentIndex = 0; }
                return _currentIndex;
            }
            set { _currentIndex = value; }
        }

        public T MoveNext
        {
            get { _currentIndex++; return this[CurrentIndex]; }
        }

        public T MovePrevious
        {
            get { _currentIndex--; return this[CurrentIndex]; }
        }

        public T Current
        {
            get { return this[CurrentIndex]; }
        }
    }

Using this becomes quite easy

 NavigationList<string> n = new NavigationList<string>();
            n.Add("A");
            n.Add("B");
            n.Add("C");
            n.Add("D");
            Assert.AreEqual(n.Current, "A");
            Assert.AreEqual(n.MoveNext, "B");
            Assert.AreEqual(n.MovePrevious, "A");

Using LINQ in one line and with circular search:

Next of

YourList.SkipWhile(x => x != NextOfThisValue).Skip(1).DefaultIfEmpty( YourList[0] ).FirstOrDefault();

Previous of

YourList.TakeWhile(x => x != PrevOfThisValue).DefaultIfEmpty( YourList[YourList.Count-1]).LastOrDefault();

This is a working example (link to the fiddle)

    List<string> fruits = new List<string> {"apple", "banana", "orange", "raspberry", "kiwi"};
    string NextOf = "orange";
    string NextOfIs;

    NextOfIs = fruits.SkipWhile(x => x!=NextOf).Skip(1).DefaultIfEmpty(fruits[0]).FirstOrDefault();
    Console.WriteLine("The next of " + NextOf + " is " + NextOfIs);

    NextOf = "kiwi";
    NextOfIs = fruits.SkipWhile(x => x!=NextOf).Skip(1).DefaultIfEmpty(fruits[0]).FirstOrDefault();
    Console.WriteLine("The next of " + NextOf + " is " + NextOfIs);

    string PrevOf = "orange";
    string PrevOfIs;

    PrevOfIs = fruits.TakeWhile(x => x!=PrevOf).DefaultIfEmpty(fruits[fruits.Count-1]).LastOrDefault();
    Console.WriteLine("The prev of " + PrevOf + " is " + PrevOfIs);

    PrevOf = "apple";
    PrevOfIs = fruits.TakeWhile(x => x!=PrevOf).DefaultIfEmpty(fruits[fruits.Count-1]).LastOrDefault();
    Console.WriteLine("The prev of " + PrevOf + " is " + PrevOfIs);

You can use indexer to get element at desired index. Adding one to index will get you next and subtracting one from index will give you previous element.

int index = 4; 
int prev = list[index-1];
int next = list[index+1];

You will have to check if next and previous index exists other wise you will get IndexOutOfRangeException exception. As List is zero based index so first element will have index 0 and second will have 1 and so on.

if(index - 1 > -1)
   prev = list[index-1];
if(index + 1 < list.Length)
   next = list[index+1];