[SalesForce] how to auto populate lookup field value based on record type

Just create a trigger on OrderItem__c. Since the shipping customer number is unique, when an OrderItem__c record is updated, you can query for the Contact whose Shipping Customer Number matches the new value of the OrderItem__c, and set the Contact lookup on OrderItem__c to match that queried Contact.

You can find information on developing triggers here.

I didn't attempt to compile this and there might be a more elegant way to write it, but it's bulkified and is generally what the logic should be:

trigger SetContactOnOrderItem on OrderItem__c (before update, before insert) {
    Set<Integer> shippingNumbers = new Set<Integer>();

    for (OrderItem__c collectNumFromOrder : Trigger.new) {
        shippingNumbers.add(collectNumFromOrder.Shipping_Customer_Number__c);
    }

    List<Contact> contactList = [SELECT id, Shipping_Customer_Number__c FROM Contact WHERE Shipping_Customer_Number__c IN :shippingNumbers];

    Map<Integer, Contact> shippingNumToContactMap = new Map<Integer, Contact>();

    for (Contact c : contactList) {
        shippingNumToContactMap.put(c.Shipping_Customer_Number__c, c);
    }

    for (OrderItem__c o : Trigger.new) {
        if (o.Shipping_Customer_Number__c != null) {
            o.Contact__c = shippingNumToContactMap.get(o.Shipping_Customer_Number__c).id;
        }
        else {
            o.Contact__c = null;
        }
    }
}  

You can use an if condition if(trigger.oldmap.get(opp.Id).RSM_Shipping_Contact__c != opp.RSM_Shipping_Contact__c)

This will make sure that code is executed only if contact lookup is changed.

 if(trigger.isbefore && trigger.isinsert)
{
  Set<ID> ConIds = new Set<ID>();

for(Opportunity opp : trigger.new)
{
      ConIds.add(opp.RSM_Shipping_Contact__c);    
}

list <contact> conlist = [SELECT Email,Id,MailingCity,MailingCountry,MailingPostalCode,MailingState,MailingStreet,Phone FROM Contact where id IN:ConIds];

 MAP<ID ,contact> mapCon = new MAP<ID ,Contact>();
 for(Contact c : conlist)
 {
    mapcon.put(c.id,c);
 }

 for(Opportunity opp : trigger.new)
 {
  if(trigger.oldmap.get(opp.Id).RSM_Shipping_Contact__c != opp.RSM_Shipping_Contact__c){
    if(opp.RSM_Shipping_Contact__c!=null)
    {
      if(mapcon.containskey(opp.RSM_Shipping_Contact__c))
      {
        contact c = mapcon.get(opp.RSM_Shipping_Contact__c);
        opp.Shipping_Street__c = c.MailingStreet;
        opp.Shipping_City__c = c.MailingCity;
        opp.Shipping_State__c = c.MailingState;
        opp.Shipping_Country__c= c.MailingCountry;
        opp.Shipping_postal_code__c = c.MailingPostalCode;
        opp.Shipping_Email__c = c.Email;
        opp.Shipping_Phone__c = c.phone;
      }
    }

    else
    {

      opp.Shipping_Street__c = null;
      opp.Shipping_City__c = null;
      opp.Shipping_State__c = null;
      opp.Shipping_Country__c= null;
      opp.Shipping_postal_code__c = null;
      opp.Shipping_Email__c = null;
      opp.Shipping_Phone__c = null;

    }
 }
 }
}