Edinburgh travelled to Dublin with a starting XV unchanged following the 19-14 win over four-time European champions Toulouse in the last round on April 7. The only switch in the 23-man squad was on the bench, where back-row Stuart McInally (hamstring) failed a fitness test, leading to a call-up for lock Steven Turnbull.
"The size of the occasion and the backing from fans has not been lost on the players," Bradley said. "Tomorrow will be a huge test. The hype, the intensity, the crowd, and the fact that we're away from home have a bearing on the size of tomorrow's challenge."
Four thousand fans are anticipated to travel as Edinburgh go in search of a place in the May 19 Heineken Cup final at Twickenham against either Clermont Auvergne or defending champions Leinster, who meet on Sunday.
Bradley added: "Ulster are one of the in-form sides in Europe and will take great confidence from their two league wins over us already this season, on top of their away win over Munster in the quarter-final.
"Our form in this competition is such that we can rightly approach this match with a lot of confidence. The character and resolve of the players has been well tested in this competition and we have prevailed.
"It's a one-off game and the underdog tag is one we wear well as a team. Edinburgh Rugby have shown that we are a force to be reckoned with on big occasions and are relishing our first semi-final in this competition."
There was no place for Chris Paterson, Scotland's most capped player and record points scorer, who is set to travel with the squad and is poised to be on kicking tee duty once again.
"If we can play close to our best, we're well capable of beating them."
Chris Henry is missing for Ulster through injury with Paddy Jackson starting at fly-half.
Henry has been ruled after failing to recover from an ankle sprain picked up in the match against Leinster last weekend. Other injury concerns Paddy Wallace, Pedrie Wannenburg and Paul Marshall have all been passed fit, with the latter named among the replacements.
Forward trio Stephen Ferris, Dan Tuohy and Rory Best have all returned to the starting XV after being rested last weekend. Ferris is a direct replacement for Henry so Willie Faloon retains his position at openside flanker.
Nigel Brady drops to the bench for the return of Best, while Lewis Stevenson makes way for Tuohy. After impressing last week in his first game in five months, Declan Fitzpatrick will start again at tighthead prop.
Brian McLaughlin has named an unchanged backline for Saturday's crucial game, meaning Jackson gets the nod over Ian Humphreys in the number 10 jersey.
Ulster: 15 Stefan Terblanche 14 Andrew Trimble 13 Darren Cave 12 Paddy Wallace 11 Craig Gilroy 10 Paddy Jackson 9 Ruan Pienaar; 1 Tom Court 2 Rory Best 3 Declan Fitzpatrick 4 Johann Muller (capt) 5 Dan Tuohy 6 Stephen Ferris 7 Willie Faloon 8 Pedrie Wannenburg
Replacements: 16 Nigel Brady 17 Paddy McAllister 18Adam Macklin 19 Lewis Stevenson 20 Robbie Diack 21 Paul Marshall 22 Ian Humphreys 23 Adam D'Arcy
Edinburgh: 15 Tom Brown 14 Lee Jones 13 Nick De Luca 12 Matt Scott 11 Tim Visser 10 Greig Laidlaw (capt) 9 Mike Blair; 1 Allan Jacobsen 2 Ross Ford 3 Geoff Cross 4 Grant Gilchrist 5 Sean Cox 6 David Denton 7 Ross Rennie 8 Netani Talei
Replacements: 16 Andrew Kelly 17 Kyle Traynor 18 Jack Gilding 19 Steve Turnbull 20 Roddy Grant 21 Chris Leck 22 Phil Godman 23 Jim Thompson