Should I do ab work at the end of my workout so, I won't hurt my back?

Ab work when done correctly should ultimately help your back.  I recommend that my clients begin their routines with foam rolling, stretching and then core (ab work) before getting into the more serious lifts.  Meet with a fitness professional to review your program and technique to make sure you are getting the most benefit from your core training.
To the contrary, you should focus on your core early in your session as you will need these stabilizers to support you regardless of the body part you are training. Performing a few core exercises early in your workout will allow other muscle groups to operate more efficiently and enhance the overall experience.

Actually, we strongly recommend performing some "core" work near the beginning of your workout after stretching and a light aerobic warm-up. This method prepares your core for the rest of your workout. Regardless of what body part you are training (chest, back, legs etc.) or what type of exercise you're performing, your core plays a pivotal role to help you maintain proper posture and protect your spine. Performing a few core exercises (for example 1-3 exercises) will help "wake up" your nervous system and enable you to contract your core muscles (abdominals, low back, and gluteals) forcefully during your workout. When your core muscles are operating optimally, you're able to move more efficiently, enhance your workout, and prevent injury. It is important to note that you should not perform a high volume of core exercises that fatigue your core. The object is to simply "wake up" these muscles.

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Important: This content reflects information from various individuals and organizations and may offer alternative or opposing points of view. It should not be used for medical advice, diagnosis or treatment. As always, you should consult with your healthcare provider about your specific health needs.