Improve SQL Performance with LIKE '%ABC%' QUERY
Without full-text search, no, there's no magic to making string parsing faster within SQL Server, other than pre-calculating results or throwing more resources at the problem.

If you have a narrow set of search patterns that are repeated over and over again, it's possible you could maintain a skinnier materialized portion of the table that meet those criteria (e.g. a table of just the PK columns representing the rows in the main table that match '%ABC%' - you could maintain these through triggers). This will reduce the amount of reads required, but may not have a serious impact on duration.

If people are entering arbitrary search strings in a non-repeatable and unpredictable way, that may not help anyway.

10 seconds seems like a long time for a table with 160K rows. If you are on V12 (and can run this query in relative isolation), you should be able to determine the waits that changed during that query, using sys.dm_db_wait_stats - it may be that you can't keep 300MB of data in memory and the wait time is all disk churn. In this case it may just be that you are sharing an overwhelmed server, so one consideration would be to move up to a better tier that provides for better performance.

Another option you can consider is application-side caching (think memcached, redis, etc.), where you have a copy of the data in your application's memory, and perform the searching there instead of within SQL Server.