Vodafone cited his experience in running capital intensive infrastructure projects as well as service-based industries.
It also said he had strong political and business connections in Europe and was used to dealing with companies with varying stakeholder interests to manage.
Vodafone is assembling an independent board for Vantage Towers as it prepares the ground for its giant flotation.
The company's shares have underperformed for years and the flotation is hoped to reinvigorate the price.
Vantage's team are expected to be urging fund managers to invest in the IPO to capture profits from future growth from 5G rollout as big telecoms operators use its masts to put up their infrastructure. They are also expected to pitch it more generally as an investment play in the growth of data usage.
Vantage Towers chief executive Vivek Badrinath said: "I look forward to working with [Grube] and the wider supervisory board to drive our strategy to enable Europe's digital transformation and to benefit from the attractive long-term trends delivering growth and value opportunities across each of our markets."
Grube said he looked forward to "delivering a successful IPO".
Vodafone will retain a majority stake in the business after the flotation, which Barclays analysts have valued at about e21 billion based on the value of Italian towers company Inwit.
Towers businesses such as Inwit and Spain's Cellnex have increased in value over the past 18 months, encouraging big telecoms companies to sell.
Vodafone chose Frankfurt over London for the float during the summer. It was seen as a blow for the London Stock Exchange but Vodafone chief executive Nick Read said the decision had been made because the lion's share of the masts were in Germany so it would attract more investors there.
Vodafone hired long-term Heineken chief executive Jean-Francois Van Boxmeer as its group chairman in May.
