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When you enter this Gym, your sunglasses-wearing pal from the Striaton City Gym will bestow another Fresh Water item on you. This recovery item heals your Pokemon for 50 points of damage.
The Nacrene City Gym focuses on Normal type Pokemon. If you've captured any wild Fighting type Pokemon, they will be super effective against Nacrene City Gym's Pokemon. If not, most other Pokemon types will handle themselves capably in the upcoming battles, you just won't gain any significant advantage.
To gain entry to the Gym Leader's chamber you must first correctly answer a series of quiz questions. First, initiate combat with the trainer School Kid Carter, who uses a level 17 Patrat. Once defeated he will direct you to read a specific book. Read it then approach School Kid Lydia. Answer "no" to her question, and then head for Scientist Satomi and answer "yes" when prompted. You'll now fight Scientist Satomi's level 17 Herdier. Satomi will stand aside once beaten to grant access to another book.
Read the second book then head back to School Kid Lydia. Respond "no" to her question and she will direct you to a third book. This book will give you a set of directions to follow that ultimately leads you right back to Lydia. This time answer "yes" to her question to start a battle against her three level 15 Lillipups. Lydia will step aside once defeated to reveal a final book. Read it to open a secret staircase down to the Gym Leader's chamber.
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Pokemon Black and White Nacrene City Gym Guide - Gym Leader
Gym Leader Lenora's first Pokemon is a level 18 Herdier. This pup is already pretty tough by default, but to make matters worse it possesses the Intimidate ability, which lowers your Pokemon's Defense. That combined with the devastating Take Down move means any Pokemon with a low Defense statistic will be knocked out quickly. Bring out your most damage-resistant Pokemon to deal with the Herdier and heal using recovery items when necessary.
Lenora's second Pokemon is a level 20 Watchog, the evolved form of Patrat. Watchog can use the Retaliate attack to brutally wallop your Pokemon when it is first dispatched. Retaliate is an attack that doubles in effectiveness if a teammate was defeated during the previous turn. This isn't the only trick up its sleeve however, as it can also use Hypnosis to put your Pokemon to sleep. Surviving the initial onslaught from Retaliate is the key to winning this battle. If you can withstand that lethal attack then it is just a matter of trading blow for blow until Watchog faints.
Gym Leader Lenora also has a Super Potion in her inventory that can heal one of her Pokemon for 50 hit points.
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Pokemon Black and White Nacrene City Gym Guide - Reward
Once you've defeated Lenora you'll be awarded the Basic Badge. This Badge ensures that all Pokemon level 30 and below will obey your commands without question. You'll also receive TM67, a Technical Machine that teaches the Retaliate skill.
All screenshots/references from Nintendo's Pokemon Black and White and author's own experience.