Solr is capable of all those things but to adequately explain how to do each of time an answer would become a mini-manual for Solr.
I'd suggest you read the actual manual and tutorials linked from the Solr homepage.
In short though:
Startswith can be implemented using Lucene wildcards.
Exact matches will only be found if a field is not tokanized. I.e. the entire field is viewed as a single token.
Contain is the default search format. I.e. a search for "John" will find any document's whose search field contains the value "John". Prefixing with - (e.g. "-John" will only find documents that do not contain John).
Ranges (be they date or integer) are possible and quite powerful, example date:[* TO NOW] would find any document whose date is not in the future.