Generic SObject update pattern

Once you have the sObjectType you should be able to construct the sObject instance using the sObjectType.newSObject(ID Id) method. You can get the sObjectType from the Id using the Id.getSObjectType() method;

Map<Id, String> idsToUpdate = new Map<Id, String>();

// Put the Id's and associated name values in the map

List<SObject> sObjectsToUpdate = new List<SObject>();

for (Id idToUpdate : idsToUpdate.keySet()) {
    SObject o1 = idToUpdate.getSObjectType().newSObject(idToUpdate);
    // Set the Name field dynamically
    o1.put('Name', idsToUpdate.get(idToUpdate));
    sObjectsToUpdate.add(o1);
}

update sObjectsToUpdate;

Here's a link to a post that I think answers your question: Infering sObject Type from Id or collection of Id's

Essentially, the solution entails:

1. Pulling the sObject "prefix" (the first 3 characters) from the id you receive
2. Get the sObjectType so you can cast your objects as the corresponding sObject (or list<yourSobject__c>)
3. Upsert the typed-records

Link to SF Docs

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Here's a Utility Method I wrote that writes a query to select ALL fields from an sObject by only giving the string value of the Object Name. It exemplifies finding an sObject's metadata (in this case, its fields).

public static string FormatFieldsForQuery(string objectName,string prefix){
    if(string.isNotBlank(prefix)){
        if(prefix.endsWith('__c'))
            prefix=prefix.replace('__c','__r.');   
        if(!prefix.endsWith('.'))
            prefix+='.';
    }
    string fields = '';
    sObjectType objectType=Schema.getGlobalDescribe().get(objectName);
    if(objectType==null)
        return fields;
    for(string f :objectType.getDescribe().fields.getMap().keySet())
        fields+=prefix+f+',';
    return fields.removeEnd(',');  
}//END FormatFieldsForQuery(string objectName,string prefix)