Colt made a real sturdy gun back then. Also, Police issued weapons were not noted for being fired a lot. Qualification and Training time mostly.
I would first check the area ahead of the cylinder. Look for excessive gap between the back of the barrel and the front of the cylinder. .006-.010" clearance is normal, more may indicate wear or damage. Holding the cylinder LIGHTLY, pull the hammer back. The cylinder should advance and the detend drop in place when the hammer is fully back and cocked. Release the cylinder and note if it rotates back and if so, how much. A minute amount is OK and lots indicates it is out of time.
If it passes the mechanical checks give it a good cleaning with old fashioned Hoppe's.
Don't use any Lead Removal Wipes on the bluing or it will remove it.
Most 20th Century .38s, especially a Colt, are capable of firing standard load .38 Spl. rounds. I would avoid +P loads as they are high pressure and could cause wear.
Take care of it and it will be shooting strong at its 100 year anniversary.