The one-time Ibrox shot-stopper was stretchered off during the second half of the dramatic 2-1 victory and replaced by Scott Gallacher after colliding with team-mate Alim Ozturk.
Alexander, a summer signing from Crystal Palace, was taken straight to hospital where doctors corroborated Hearts’ own fears that the player had sustained a serious facial injury.
The 36-year-old will sit out this weekend’s visit of city rivals Hibs and is also likely to miss the Petrofac Training Cup tie at Livingston, a trip to Raith Rovers, the League Cup clash with Stenhousemuir and the visit of Falkirk on 30 August.
With Scotland in Euro 2016 qualifying action in the first weekend of next month, Alexander, who also coaches the club’s goalkeepers, could return for the 13 September outing at Dumbarton.
Neilson, who looks set to recall ’keeper Jack Hamilton from his loan stint at Stenhousemuir as cover, said: “Neil took a bad one. He’s got a fractured cheekbone so he’ll be out for around four weeks.
“It’s not great but we’ve got a squad and I thought that Scott Gallacher came on on Sunday and did very well. He’s a good ’keeper and that’s the reason that we brought him in. I’m really disappointed for Neil but he’s had a fantastic game for us. He’s a leader about the place as well.”
Neilson, however, reported that striker Soufian El Hassnaoui began training at the weekend after undergoing a groin operation earlier this summer.
The Moroccan signed from Dutch side Sparta Rotterdam on a three-year deal during the close season but is yet to kick a ball for the Gorgie outfit. Neilson said: “We’ve still got a couple of injuries but Soufian trained over the weekend for the first time. He’ll join the squad in the middle of the week.”
On defender Callum Paterson, who underwent surgery to repair cruciate knee ligament damage after picking up the injury in last month’s Petrofac Training Cup win over Annan, Neilson added: “Callum has his knee problem but that should be another three or four weeks hopefully before he’s back training with the group.”
The Tynecastle outfit have also announced that they will be roared on by a capacity home crowd for this Sunday’s visit of arch rivals Hibs.
The last few remaining tickets were sold out within ten minutes of going on sale yesterday morning.
Hearts were backed by 1,700 fans at Ibrox and have sold over 12,000 season tickets for the campaign.
Meanwhile, former Hearts captain Michael Stewart insists Charles Green’s reappearance is not an “unwanted sideshow” for the Rangers players but instead a smokescreen for manager Ally McCoist.
Stewart has been critical of McCoist’s management at Ibrox in recent seasons and believes Sunday’s 2-1 defeat by Hearts in their opening game of the new Championship season was another example of the former Scotland striker’s deficiencies in the dugout.
In the wake of the Hearts loss, McCoist claimed some players were beginning to be affected by the continuing off-field issues, describing the re-emergence of former chief executive Green as a possible future investor in the club again as “a little bit of a sideshow we can do without”.
The Ibrox boss also revealed “one or two” players had spoken to him about the “timing” of the boardroom struggles being played out so publicly.
However, Stewart, who spent three years at Hearts during Vladimir Romanov’s turbulent reign at Tynecastle, is adamant “a good manager” should be able to shield non-football concerns from his players and believes the Light Blues’ squad is experienced enough to cope.
He said: “It was the same thing last year when Charles Green reared his head again. Ally talked about the same thing, a sideshow he could do without.
“These things are not going to affect the players. I had it myself when I was at Hearts. Obviously there were lots of things that happened off the pitch.
“But a good manager protects his players and makes them comfortable with what’s happening on the football side, and you ignore all that sort of stuff. So, I don’t think that had any effect on these players.
“All you have to do is look at this Rangers team. It’s full of experienced players, it’s not a young team, there’s no youngsters there, so these guys are not going to worry about that.”
Instead, Stewart believes Sunday’s result, which came just days after Rangers’ unconvincing extra-time victory over ten-man Hibs in the Petrofac Training Cup, was down to deficiencies at Rangers and the progress made at old club Hearts.
He said: “I think a vast improvement has been made with Hearts in a short period of time. They’ve obviously been able to recruit players after the transfer embargo was lifted but there’s a huge improvement.
“You can see the style of play they’re wanting to implement, you could see that on the pitch quite clearly on Sunday.”