When you say you bought an old domain, do you mean you bought one from a person/business/website that never got dropped/deleted/etc.?
Because if you bought one that was dropped/etc. and got it for registration fee then it's life basically starts out new again (for the most part) and it's considered a new domain for all intents and purposes. I buy dropped domains and have for a very long time (as well as new domains) and right after I get them I will slap up a temporary/coming soon page/site and they always get indexed right away, within less than an hour.
If you bought a domain before it was dropped/etc. then what was on the domain as far as website if anything can play a role in how fast it gets crawled again. If it was a highly inactive domain/site and you bought it, the SE's wouldn't just think it's not very active and won't crawl it that often.