That Unity configuration looks overly complex, the Ninject configuration is much simpler. I'm not sure why that Unity configuration is that complex.
I grabbed the Ninject configuration and translated to Simple Injector. Since I'm not familiar with Solr, it could be that Solr depends a few extra tweaks in the Simple Injector configuration, but I think this will get you going:
// using SimpleInjector;
public static void Configure(Container container, string serverURL)
{
container.RegisterInstance<IReadOnlyMappingManager>(
new MemoizingMappingManager(new AttributesMappingManager()));
container.Register<ISolrDocumentPropertyVisitor, DefaultDocumentVisitor>();
container.Register<ISolrFieldParser, DefaultFieldParser>();
container.Register(typeof(ISolrDocumentActivator<>), typeof(SolrDocumentActivator<>));
container.Register(typeof(ISolrDocumentResponseParser<>), typeof(SolrDocumentResponseParser<>));
container.Register<ISolrFieldSerializer, DefaultFieldSerializer>();
container.Register<ISolrQuerySerializer, DefaultQuerySerializer>();
container.Register<ISolrFacetQuerySerializer, DefaultFacetQuerySerializer>();
container.Collection.Register(typeof(ISolrResponseParser<>), new[] {
typeof(ResultsResponseParser<>),
typeof(HeaderResponseParser<>),
typeof(FacetsResponseParser<>),
typeof(HighlightingResponseParser<>),
typeof(MoreLikeThisResponseParser<>),
typeof(SpellCheckResponseParser<>),
typeof(StatsResponseParser<>),
typeof(CollapseResponseParser<>)
});
container.Register<ISolrHeaderResponseParser, HeaderResponseParser<string>>();
container.Collection.Register<IValidationRule>(new[] {
typeof(MappedPropertiesIsInSolrSchemaRule),
typeof(RequiredFieldsAreMappedRule),
typeof(UniqueKeyMatchesMappingRule)
});
container.RegisterInstance<ISolrConnection>(new SolrConnection(this.serverURL));
container.Register(typeof(ISolrQueryResultParser<>), typeof(SolrQueryResultParser<>));
container.Register(typeof(ISolrQueryExecuter<>), typeof(SolrQueryExecuter<>));
container.Register(typeof(ISolrDocumentSerializer<>), typeof(SolrDocumentSerializer<>));
container.Register(typeof(ISolrBasicOperations<>), typeof(SolrBasicServer<>));
container.Register(typeof(ISolrBasicReadOnlyOperations<>), typeof(SolrBasicServer<>));
container.Register(typeof(ISolrOperations<>), typeof(SolrServer<>));
container.Register(typeof(ISolrReadOnlyOperations<>), typeof(SolrServer<>));
container.Register<ISolrSchemaParser, SolrSchemaParser>();
container.Register<IMappingValidator, MappingValidator>();
}