While you may be interested in participating in these events "just for fun," it is important to understand that your Borzoi will be encouraged to run, turn, and stop. At 100%. If he's not in shape, at best he won't be performing as well as he could. And at worst, he'll injure himself.
So with disclaimer duly noted*, here are some general suggestions:
I certainly recommend that you get your veterinarian's opinion on the condition of your dog, and whether there are any medical reasons your dog should not participate in performance events. A dog under the age of one is typically not allowed to participate in events, though shorter practice runs might be allowed. An older dog might be willing to run, but common sense might suggest otherwise.
Just like a human, a dog isn't going to perform well if he isn't in shape. Borzoi certainly have their "couch potato time" but will also spend short periods of time running hard, given the chance. While we can't give them Border Collie energy, we can certainly give them more exercise. It's not particularly difficult:
Depending on what type event you're attending, your dog will get a varying amount of exercise. A LGRA straight race is 200 yards, for example. That should not be an inordinate distance, so while your dog will be panting, he should not be totally drained. On the other hand, at an open field coursing event your dog might run a far longer distance - a mile easily. So you can expect him to be more tired, and he'll need longer to cool down after the course. If your dog appears more tired than you'd expect, or is not recovering after a reasonable period of time - pull him out of the event right away, and consider medical attention.
Injuries are also possible at performance events, most commonly to your dog's feet. So check them after each run, check for cut pads, check for stickers in between toes, etc. Certainly watch your dog to make sure he's not limping or otherwise acting injured. Again, pull him out of the event and consider medical help if something doesn't seem right.
In all events, your dog should be "walked out" after each run, typically 5-10 minutes of walking, rather than putting him directly in a crate or pen.
You certainly don't want your hound running on a full stomach, but you do want him to have some energy for the day. I offer a *very* light meal (normal kibble) early in the morning of an event, assuming there will be a couple hours before the dog is actually running. I say "offer" since my dogs are usually quite anxious in the morning of an event, and will frequently not eat at all in the morning. Take along some dog biscuits and give him a couple after each run during the event. And then feed him well when you get home at the end of the day, he's likely to be tired and hungry.
Your Borzoi will need water during the entire day, whether he's running or not. Bring enough with you, in case the locale does not offer water, and ensure that your dog has plenty available at all times. It's not unusual for a dog to drink 3-4 quarts of water during the day.
sportsvet.com offers some great information from Dr. Robert L. Gillette on Canine Sports Medicine and Optimizing Performance.
* I am not a veterinarian, nor any kind of "health expert" - I simply participate in the events and try to keep my dogs in shape for them. Consult your vet if you have any questions about participating in performance events with your dog.