I've solo'd a prime many times, and in fact, it's my main source of income.
1) Get perpetuation costs as low as you can get. You'll be fighting purely with your avatar, and when that avatar goes, you're dead. Keep in mind that you'll have no time to rest the entire time, so, you need to keep up MP at all times.
2) Use magic attacks over physical ones. All the primes seem to have a 50% resist physical damage trait, so using a physical bloodpact will have less effect than a magical bloodpact. Using the merit bloodpacts will have the greatest effect in terms of damage.
3) Allow your avatar to get TP. As a general rule, I usually wait for my avatar to get around 80% TP before using a bloodpact. This allows for more damage efficiency.
4) Use Elemental Syphon and Sublimation as often as possible. Remember point one where I said that MP needs to be kept up at all times. However, when using elemental syphon, use the double weather to your advantage. For example, when facing Shiva, use the Ice Elemental to get more MP out.
5) Above all else, do not panic. Always keep a cool head, especially if something unexpected happens. And never be afraid to leave if things turn especially dangerous. Sometimes, it's better to run away to fight again in five minutes than be killed and have to home point.
Wish you the best of luck!