Baker, who has held the role with the Sky Bet League Two side since 1997, is a lifelong Hammers fan and regularly makes the trip from the Cotswolds to watch West Ham.
And he admits being in the boardroom, where he will wear the scarves of both clubs, rather than the stands will make this second-round clash a memorable evening, whatever the result.
"It's an unbelievable draw, it really is, for not just me, but for the whole club," he told the Gloucestershire Echo.
"I used to watch West Ham regularly in my 20s and I even went to Brussels to watch them in the European Cup Winners' Cup final against Anderlecht in 1976, which they lost 4-2.
"During my formative years West Ham had so many great players and for a Cheltenham lad to be travelling to the East End of London was such a contrast
"The atmosphere at Upton Park is always good with the crowd virtually on the pitch and it will bring back memories for me of roasted peanuts and jellied eels outside the ground.
"I've never been in the boardroom before so I am looking forward to going there with my hometown club and it'll be a proud moment and another great day for Cheltenham."
Tackling top-flight opponents is something Mark Yates' men have become used to having taken on both Tottenham and Everton in the FA Cup in recent seasons and this will be their third Capital One Cup clash against Premier League opponents in the last eight years.
The West Ham faithful are targeting this competition as a way to end their 33 year-long wait for a trophy and plenty of them will be present to see their team host a side from the fourth tier of English football after a heavy reduction in ticket prices.
This will be their first competitive meeting with the Hammers, with the clubs having done battle three times in friendly matches between 1947 and 1952.
The Cheltenham class of 2013, losing play-off semi-finalists last season, have suffered something of a hangover during the opening weeks of the new campaign.
But they picked up their first win of the new campaign on Saturday, a 1-0 victory at Accrington thanks to Jason Taylor's goal, and will head to East London in good spirits.
They will be without experienced striker Jamie Cureton on Tuesday evening with a dislocated shoulder but fellow forward Terry Gornell has declared himself fully fit to make the trip.
- Sports & Recreation
- West Ham